Gruyere is DELICIOUS.
It melts without getting too gooey or soupy, and it is the traditional cheese component of a Croque Monsieur (if you're into that ham thing...).
You could actually probably use any of the cheeses in that "variations" list, but I love Gruyere so that's my recommendation.
It's widely available but tends to be kind of pricey here in the US.
Brie is also a good choice; since it is soft to begin with, it melts nicely.
(Just make sure to cut off the rind before putting in the sandwich - that would be a weird texture combo.)
It's great in grilled or toasted cheese sandwiches because of the richness and slightly buttery flavor.
As a local reference, check out Gorilla Cheese's menu for some tasty ideas; they do classic grilled cheeses with cheddar, American, gruyere or mozzarella, but they make others that have additions of other non-cheese components.
cooking.stackexchange.com/users/6808/Laura
I'm not sure if this counts as 'natural' - but you can transform nearly any cheese into a melting cheese transform nearly any cheese (a better version!)
into a melting cheese .
Wondra flour and a little cream go in with your crumbled/shredded cheese into the steamer and steam till its gooey - it forms a stable emulsion.
Then you can pour and cool it into slice, a burger, or into a delicious toasted cheese sandwich.
Other natural good choices include Gruyere and Comte.
cooking.stackexchange.com/users/1374/rfusca
Provolone and mozzarella fit your spec, although I find them frankly too mild for grilled cheese.
Brie separates slightly, but is otherwise excellent in grilled cheese (particularly if you add sweet notes to the dish.
I've not tried Camembert, but it should work about as well as Brie, I would think, and be slightly more flavorful.
cooking.stackexchange.com/users/3819/Dave Griffith
No matter what cheese you end up using (cheddar with a little mustard is my favorite), if it is semi-hard like cheddar or provolone, it will melt more evenly if it is grated.
cooking.stackexchange.com/users/6688/AaronN
colby cheese?
Wisconsin cheddar?
i always like pepperjack, but if you want a mild taste that may not work (its slightly zingy)
cooking.stackexchange.com/users/5509/zompz
Mimolette is my favorite.
It melts extremely well and has a nutty enough taste that it complements other flavors quite nicely, rather than just adding texture or fat/calories. :-)
It's also great on its own in a toasted cheese sandwich.
Of course, the appearance, the story about its appearance, and its history are all nice as well if you're in a pedagogical mood when presenting it.
cooking.stackexchange.com/users/7481/Art Taylor
My favorite combo is swiss and american.
I know you said no processed cheeses but those two compliment each other really well.
A lil pepper and mustard and hmmm thats damn good.
cooking.stackexchange.com/users/7278/riotburn
It shouldn't be a surprise, but, the fattier the cheese, the better it melts.
Cheddar is a good example.
cooking.stackexchange.com/users/7425/mines
smoked Havarti and marbled cheddar are great for grilled cheese sandwiches
cooking.stackexchange.com/users/35253/user35253
My kids love a take on aa sandwich they saw on food network.
When you butter the bread dip it in some fresh grated parmasean.
And in the sandwich.
Lotsa cheese american swiss mozzerella monterey jack a slice of each!
Enjoy
cooking.stackexchange.com/users/11087/SerraRain
Manchego.
Went to a nice restaurant in San Diego, and as an appetizer they served grilled cheese sandwiches with (what I think was) creamy vodka sauce for dipping.
The sandwiches used Manchego cheese on Sourdough bread.
And they used truffle butter, but I haven't been able to find that at the store.
But in recreating it, the sourdough/manchego dipped in vodka marinara sauce is the best grilled cheese sandwich I've had, and it's wonderfully simple.
cooking.stackexchange.com/users/23736/Tim
I enjoy both toasted and grilled cheese sandwiches, but I generally only alternate between havarti and muenster cheese. I would like to branch out but don't know where to start. These are the factors I think are important in melted cheese sandwiches:   Melt well and fairly quickly  Fairly mild taste and texture  Generally available  Be a natural, dairy cheese (aka not Cheeze Whiz or American cheese)    What cheeses fit these requirements? Are there other components that are vital to the cheese element of melted cheese sandwiches?
What cheeses work 'best' in melted cheese sandwich applications?
cooking.stackexchange.com
<cheese><sandwich>
cooking.stackexchange.com/questions/17885
cooking.stackexchange.com/users/7082/Katey HW
Gruyere and Comte are good options, while Provolone and Mozzarella might be too mild to work well. Semi-hard cheeses like cheddar can be grated for more even melting and soft cheeses like Brie or Camembert should also work. Other popular options include Mimolette and Pepper Jack. You can use gruyère, mozzarella, cheddar or comté.
You can use gruyère, mozzarella, provolone, cheddar or comté. Try using camembert, brie or mimolette.