The chopped cheese is decided on a flat-top griddle, and the verb in the name is the whole technique. Ground beef goes onto the hot steel, and as it cooks it is chopped and worked with the edge of a spatula against the griddle, broken down with sliced onions until the two become a single seared, crumbled mass rather than a patty. Slices of American cheese are laid over the pile while it is still on the griddle so they melt into it, not onto it, and the cheese binds the beef and onion into one molten layer. That griddle-chopping is what separates it from a burger and from a cheesesteak both: the meat is neither a formed patty nor thin-sliced steak, it is ground beef worked loose and fused with cheese on the steel.
The craft is in the chop and the roll. Worked hard against the hot flat-top, the beef develops a deep, even sear across far more surface than a patty would, and the onions half-fry into it so their sweetness is inside the meat rather than sitting on top. The cheese is melted on the griddle and folded through, which is the move that makes the filling cohere instead of scattering. That hot, loose, fused mass is then scooped onto a soft hero roll, the same long roll the sub family runs on, chosen because it is sturdy enough to carry a heavy, greasy filling without folding but soft enough not to fight it. Shredded lettuce, tomato, and mayonnaise go in cold, on top of the hot meat, and they are not garnish: they are the cool, crisp, acidic counter that keeps a roll full of seared beef and melted cheese from reading as one heavy thing. It is made to order in under two minutes on a griddle that never stops.
The variations are small and honest. The build moves with the order: extra cheese, jalapeños for heat, a fried egg laid on top, or a swap to chopped chicken or turkey on the same roll with the same method. It belongs to the long-roll family alongside the hero and the hoagie, the bodega's own griddled answer to the sub, and those relatives deserve their own articles rather than being crowded in here.