This uniquely (and quite recent) New York variation of a hamburger is an easy addition to a burger joint menu.
It can be offered as a lower cost option, even though the price of producing it is the same as a proper burger.
The chopped aspect is simply that the formed burger patty is chopped up while being grilled. A little bit different to Sloppy Joes which are unformed minced beef from start to finish.
The cheese can lie on top, or be chopped in – it doesn’t seem to matter.
The bun is a sub by necessity – if it is a burger bun there is too much room for the meat to slip out the side. Even with a half-open roll, the Chopped Cheese is wrapped in foil or grease-proof paper to keep everything in place.