The Jucy Lucy is a cheeseburger built inside out, and the inversion is the entire sandwich. Instead of laying cheese on top of a cooked patty, the cheese is sealed inside two patties pressed together, so it melts as the burger cooks and arrives as a molten core rather than a draped slice. The defining element is that hidden, liquid center. Everything else, the soft bun, the griddled onions, the standard burger frame, is conventional; the one radical decision is moving the cheese to the inside, where heat turns it into a pocket of lava that defines both how the burger is made and how it has to be eaten.
The craft is a sealing problem disguised as a burger. Two thin beef patties are formed around a slug of cheese and crimped at the edges so the cheese cannot escape during the cook, because a weak seal blows out on the griddle and the whole premise leaks away. The patty is cooked longer and more carefully than an open one, since the center has to reach full melt while the exterior builds a crust without burning, a slower and less forgiving cook than a standard burger. The bun is soft and yielding on purpose, a structural choice as much as a textural one, because it has to absorb the rush of molten cheese and fat the moment the patty is bitten without falling apart. The cheese itself comes out genuinely dangerous, retaining heat far longer than a melted slice on top, which is part of the sandwich's identity rather than a flaw. A bar griddle is built for exactly this: form, stuff, seal, and cook low enough that the center catches up to the crust.
The variations stay inside the stuffed-patty idea and mostly change the filling. Different cheeses behave differently in the core, and some builds seal in jalapeños, bacon, or other additions alongside the cheese, each a small swap on the same structural trick. The Jucy Lucy is one regional dialect of the American burger, which fractures into mutually incompatible methods, the smash, the steamed, the onion-fried, by region. Those are their own sandwiches with their own rules and deserve proper articles of their own rather than being crowded in here.