Well my question is exactly what is says in the title.  I come from a culture that has hamburger-style meal. Unlike hamburger, our plyeskavitsa (or pljeskavica using another type of transliteration) tends to grow in diameter when the portion size increases. That is to say, when I order a small plyeskavitsa, the patty will have (for example) diameter A and thickness B. When I order large plyeskavitsa, it will have diameter 3 A and thickness maybe 1.5 B. This way, the eating procedure is same for all sizes: You start biting at one end, you bite off a piece through the bun and the patty and basically repeat the procedure until it's consumed. If you buy a bigger portion, you'll just need to repeat the procedure more times.  On the other hand, with American burgers, when the portion size increases, the diameter will stay the same, but the thickness significantly increases as well as the number of patties (for example let's think of say BigMac or BigKingXXL). When I try to apply procedure for plyeskavitsa or say regular hamburgers, what happens is that when I bite down, the middle layers get pushed out of the sandwich, making a total mess of everything.   So my question is: How am I supposed to eat thick hamburgers without having the middle part spill out? 
A somewhat strange but effective technique is to fix the contents by piercing one or two toothpicks through the entire burger, before taking the first bites on the opposite side. When you approach the middle you can take them out, at this point the filling should tend to slip in your mouth rather than the other way. 