THREE'S COMPANY | The brain trust behind El Celler de Can Roca. From left, head chef Joan Roca, sommelier Josep Roca and pastry chef Jordi Roca. Photography by Nacho Alegre for WSJ. Magazine

WHEN YOU REACH NUMBER ONE on the influential San Pellegrino list of the World's 50 Best Restaurants, suddenly everybody wants a piece of you. For the Roca brothers, the Catalan triumvirate whose El Celler de Can Roca, in Girona just north of Barcelona, ascended to the top spot last spring—knocking Denmark's Noma down to number two—reservation requests hit the stratosphere overnight, with hundreds of calls pouring in daily for just 50 spots each at lunch and dinner. In February, sold out for the year, the restaurant opened up bookings for 2015.

A torrent of speaking and cooking invitations followed, along with lucrative inducements to open new restaurants—in New York, Singapore, Paris, Shanghai. "Every month a new proposal comes in," says Joan Roca, eldest brother and head chef, standing in the kitchen at lunch one recent afternoon. "I got an offer from London this morning, and I said, 'No, thank you.' We'll never open another El Celler de Can Roca anywhere else."

Photos: The Brothers Roca

Click to view slideshow. Photography by Nacho Alegre

You don't stay number one for long by selling your soul to the highest bidder. But all the attention, they realized, deserves a response. Joan and his brothers—Jordi on pastry, Josep on wine—devised a novel way to appease their new far-flung fans: They would take their restaurant on the road for a while.

In August, El Celler de Can Roca will embark on something called the Roca & Roll World Tour 2014. First announced on their blog in December, the tour is a three-year project with a corporate sponsor, Spanish bank BBVA. This summer they'll hit Mexico City, Lima, Medellin and, if all goes well, New York. Next year they're hoping for stops in Istanbul, Houston and maybe Santiago, Chile. With 26 cooks and waiters coming along, the restaurant will shut down for five weeks. "We prefer to be closed," says Joan, "it's more honest. If the restaurant travels, it's really the restaurant that travels." At each stop, the brothers will work closely with top local restaurants—with old friends like Enrique Olvera and Gaston Acurio as well as some new ones—tailoring their food and drink to the particular locale. Collaboration is, after all, in their professional DNA.

El Celler de Can Roca may in fact be the world's most collaborative high-end restaurant, a fraternal enterprise built on interdisciplinary cooperation. At every opportunity, the three brothers have eagerly consulted each other, along with great thinkers outside their business. Fine artists customize plates for their food; industrial designers and scientists help solve kitchen conundrums. Last spring, a special gastronomic presentation called El Somni (the Dream) made its debut in Barcelona, with contributions from a team of performers and visual artists. A documentary on the project premiered at the Berlin International Film Festival in February.

Though they are enjoying their moment in the spotlight, for years the Roca flagship was overshadowed by another sibling endeavor, Ferran and Albert Adrià's nearby El Bulli, which shuttered in 2011. "The great attention Ferran got, for a long time, allowed us to work quietly in the shadows," says Joan. "When he stopped being so visible it was our moment to shine."

In the late '80s Joan spent a month working with Ferran at El Bulli, back when both chefs were just starting out. The restaurant wasn't yet the hotbed of experimentation it would become, but, says Josep, "we knew things were happening." The sense of freedom, evident in the kitchen already, inspired an avant-garde shift at El Celler de Can Roca. By the mid-'90s both restaurants were exploring a mix of cutting-edge science and high-end gastronomy—the Rocas breaking as much new ground, in their own way, as the Adriàs were but with far less fanfare.

In 1995, Joan, with two friends from catering school in Girona, developed the Roner, the first serious professional sous-vide cooking device, based on lab equipment for keeping test tubes warm. (Sous-vide—a cooking process developed in France in the '70s—involves sealing ingredients in airtight plastic bags and submerging them in water maintained at precise temperatures.) A sous-vide cookbook followed in 2003. Looking for a way to capture pure, concentrated aromas, Joan turned an industrial distillation device—a rotary evaporator—into a customized kitchen tool that other chefs have embraced.

 Josep, meanwhile, has broken just about every wine rule there is, and made up a few of his own. He mixed two wines in the same glass to come up with the perfect pairing for a complex dessert—"if we can't find a match with one wine, why not a cocktail of wines?" he says—and once served a Pedro Ximénez stripped of its alcohol beside an eau-de-vie made from that same intense wine ("its body and soul"). Working with a chemist and a Cava producer, he developed a sparkling wine, to be poured over oysters, that's effervescent despite having a thick, sauce-like consistency.

In 2001, youngest brother Jordi, a sort of Iberian Willy Wonka, began transforming commercial perfumes—25 in all—into a series of exuberant desserts based on their grace notes. The idea had started with a passage from Patrick Süskind's novel Perfume. "He mentioned bergamot," says Jordi, "an ingredient we didn't know." A client, who sold fruit, brought in seven cases for Jordi to play with. The citrus aromas called to mind Josep's cologne, Eternity by Calvin Klein, which became the inspiration for Jordi's first perfume creation.

Two years ago he released his own fragrance, developed with a Barcelona perfumer, based on his "lemon cloud" dessert. "Anarchy," another sweet provocation, featured 50 intense disparate elements—including black olive, chocolate, coffee, yuzu, licorice, lychee—arranged with no rhyme or reason on a single plate. "It's like a sociological experiment," says Jordi. "It's insanity, a dish with endless possibilities. Everybody has a different experience eating it."

AT YOUR SERVICE | A view into the dining room at El Celler de Can Roca, which has broken as much gastronomic ground as Ferran Adrià's El Bulli. 'When Ferran stopped being so visible,' says Joan, 'it was our moment to shine.' Photography by Nacho Alegre for WSJ. Magazine

Each brother has his particular niche, but they're forever tasting, sharing, bouncing ideas off each other. "I'm lucky that my brothers are so generous," says Josep, "to be able to go into the kitchen and find someone who listens. Not everyone has that chance."

As they learned to work closely together, the Rocas had a fine example to follow: They grew up in the restaurant business, surrounded by family—aunts, uncles, grandmothers, cousins—who all lent a hand at their parents' working-class tavern, the original Can Roca. Every morning at 6:30, their father—Josep Senior, a bus driver by vocation—opened up the place, as he still does every day. Later their mother, Montserrat, would arrive in the kitchen to prepare her simple Catalan fare. Before school and after, the boys were expected to do their part.

Montserrat and Josep still keep their place going—the food and decor are mostly unchanged, but the business has been buoyed by their sons' nearby success. If you can't get into the Roca brothers' three-star Michelin restaurant, you can taste their mom's cooking around the corner—just 10 euros for a three-course meal. Most days around noon you'll find all three brothers—along with their entire staff—scarfing down lunch as I did on an overcast Friday in early January. The spread included fried sardines, cannelloni and pork ribs. After launching their own restaurant, the brothers adopted the original Can Roca as their staff canteen—giving "family meal" a whole new meaning.

FOR JOAN, who turned 50 in February, a life in the kitchen seemed almost pre-ordained. "He was the perfect son," recalls Montserrat, "always wanting to help his mother." Josep, two years younger, was far more interested in kicking a soccer ball around with his friends—he was goalie on the local team—but he had restaurant duties too. "He said he would have been a soccer player if he didn't have to help at the restaurant," whispers his mother, "but the truth is, he wasn't that good." He spent a lot of time in the dining room where patrons gave him a nickname, Pitu—a diminutive of Josep—that sticks to this day. "I was an extrovert, and a bit of a devil," he says. He was also a practical joker, roller-skating through the dining room while carrying dishes, wearing a snorkel and mask to peel a sack of onions.

Jordi, 14 years younger than Joan, was just 11 in 1986 when his brothers—finished with catering school—decided to transform the house next to Can Roca into a restaurant of their own, the Cellar of Can Roca. In 2000, Jordi joined them as partner and pastry chef. Seven years later the restaurant moved across the street into more spacious, luxurious digs, a beautiful early 20th-century home with a modern annex that's all windows and light.

In the 28 years since El Celler first launched, the brothers have never strayed far from where their story began: Joan lives above the restaurant; Josep atop its first incarnation; Jordi nearby in the place he was born. They travel frequently—when the restaurant is open, never more than one at a time—but Girona is home.

When I visited, the three men had just returned from their annual winter break—the restaurant shutters for two weeks around Christmas. Joan had been in New York, where he dined at Eleven Madison Park—"fantastic," he says—and scouted the city as a potential stop on the August world tour. Josep had taken his wife, Shani, and children, Marti and Maria, to Disneyland Paris before spending a few days in Paris where he picked up some rare Chinese tea—a passion—and, from a friend, French sweetbreads. (Joan would serve them that night as an off-menu special for a few friends of the house.) Jordi had brought his mom and dad to Mexico, to meet his new in-laws in Guadalajara—last year he married Mexican pastry chef Ale Rivas, a former stagiaire at his restaurant. Together they run Rocambolesc, a whimsical ice cream parlor, with an outpost in Girona and one on the beach along the Costa Brava.

Since winning the top spot on the 50 best list, the brothers have made a special effort to be in the restaurant together as much as possible—a lot of side projects were put on hold last spring. "Everyone comes and they want to see us," says Joan. "We have to be respectful of patrons who've made a big trip to be here. It's our responsibility." Lately, though, they've begun easing back into their more usual manic juggle. Joan has been working on a cookbook for ambitious home cooks. Josep has his own book in the works, with a Catalan psychologist, exploring his theory that "wines resemble the people who make them." And Jordi is hoping to tackle an ice cream book.

At the start of dinner service on Friday night, the kitchen swarmed with young cooks. Sous-vide baths bubbled, a wood fire crackled, a plancha sizzled, a rotary evaporator distilled the essence of dirt. El Celler's cooking combines the oldest and newest techniques into what the Rocas like to call "techno-emotional" cuisine, food that's nostalgic and cerebral at once. Near the entrance, a blackboard stood, scribbled on with brainstormed ideas for an upcoming event—small plates for a charity party sponsored by Macallan Scotch. "It's important for us to have these sorts of collaborations," said Joan. "They provide an incentive for creativity. Something will be left behind for the menu when we're done."

As I pulled up a stool at a counter, I watched the preparations for a gastronomic journey down El Celler de Can Roca's memory lane. The Classics Menu, one of two meal options available, features the restaurant's greatest hits ranging back to the era when Joan and Josep (before Jordi came on) first discovered the thrills of French nouvelle cuisine. A seminal trip to the great country restaurants across Spain's northern border—icons like Troisgros in Roanne and Pic in Valence—helped inspire upscale spins on Catalan classics, like the rich timbale of foie gras and caramelized apple (circa 1993) that kicked off my dinner that night.

The nostalgic dishes delivered a delicious history lesson, but the real fireworks—and what makes this restaurant one of the best in the world—belong to the 25 up-to-date courses on the much longer Feast Menu, a four-hour experience I was served in the dining room the following night. The menu features presentations as striking as the food itself. A black-truffle bonbon arrived inside a hollowed-out stone, while a salad of sea anemone and razor clams spilled out over the spikes of a metallic anemone. Langoustine tail, steamed table-side in sweet Palo Cortado, filled the whole restaurant with a perfumed cloud of sherry. Sourdough ice cream with fried lychees quivered on an undulating rubber pedestal—a gesture apparently intended to reference living yeast, a final dose of Jordi humor. "How does it move?" I asked my waiter. "Chinese battery," he said.

One of the first tastes of the meal arrived shrouded in an origami globe. Inside were five tiny bites, each a culinary tribute to a country the Rocas know and love: Mexico, China, Morocco, Korea, Peru. The world tour, when it's done in three years, may have hit all of those spots, and many others too. "It's a present for our employees," said Joan, "and also a way to find new ideas. It's important to change things up, to break the routine, to break the monotony."