The Juicy Lucy is the cheese-stuffed burger spelled with the i, and the spelling is not a typo or a footnote: it is the identity. Two Minneapolis bars serve the same idea, a slug of cheese sealed inside a beef patty so it melts into a molten core, and they spell it differently on purpose, one dropping the i and one keeping it, and each insists its spelling is the true one. The sandwich therefore arrives with its provenance built into its name. Order it by spelling and you have already chosen a side in a rivalry that the two kitchens treat as a matter of craft rather than branding, because each spelling also signals a slightly different patty.
The craft is a geometry problem. The cheese is enclosed between two beef portions pressed into a single patty and crimped at the rim so the molten center cannot blow out on the flat-top, and the i-spelling camp builds its patty wide and the seal broad, so the cheese spreads into a flat disc across the interior rather than pooling in a narrow well. That shape changes the eating: the core is distributed, so a bite near the edge still hits cheese, and the rim has to be sealed cleanly all the way around or the whole premise leaks onto the steel. The cook is slower and less forgiving than an open burger, because the buried cheese has to reach a full liquid melt while the exterior builds a crust without scorching, and the patty is cooked through by necessity rather than to a chosen doneness. The cheese exits genuinely dangerous, holding heat far longer than a draped slice, which both kitchens treat as the point and warn you about. The bun is soft on purpose so it absorbs the rush of fat and cheese at the first bite instead of fracturing.
The variations stay inside the stuffed-patty idea and mostly change the core. Different cheeses behave differently sealed inside, some flowing and some staying gummy, and some builds enclose jalapeño or other additions alongside the cheese, each a small swap on the same trick. The Juicy Lucy is one regional dialect of the American burger, a thing that splinters into mutually incompatible methods, the smash, the steamed, the onion-fried, by region. Those are their own sandwiches with their own rules, and each deserves its own article rather than being crowded in here.