On a chopped cheese, the variable that matters is not the beef or the cheese but where the heat goes in. The standard build is settled: ground beef chopped and seared on a flat-top, worked loose with onions, fused with melted American cheese, scooped onto a hero roll under cold lettuce, tomato, and mayonnaise. The hot-pepper build changes one thing and changes the whole balance. Cherry peppers or jalapeños are added to a sandwich whose entire structure is already a study in heavy seared beef bound by molten cheese, and the pepper is the only sharp, acidic, burning element in the assembly. That single decision is the sandwich. It is the bodega answering a customer who wants the same fused griddle mass with a counter to the fat rather than only the cool, soft toppings that the plain version leans on.
The craft is in when and where the pepper lands. Pickled cherry peppers, sliced, can go onto the griddle with the beef so the chop works their brine and a little of their burn into the meat itself, which makes the heat structural rather than a topping sitting on top. Sliced jalapeños more often go in cold with the lettuce and tomato, which keeps their bite distinct and crisp against the hot filling instead of folding it in. Either way the pepper does double work: it supplies the acid that cuts a roll full of greasy beef and melted cheese, and it supplies a burn that the mayonnaise then has to chase and round off. The roll is still the soft hero the long-roll family runs on, sturdy enough to carry the load without folding. The pepper does not change the structure of the sandwich. It changes what the sandwich tastes like at the moment the fat would otherwise take over, which is the whole reason a customer asks for it by name.
The variants are small and move with the order. Pickled cherry peppers chopped into the meat, raw jalapeño added cold, a hotter swap to long hots or banana peppers, or a doubling of cheese to push back against a build that now carries real heat. It belongs to the long-roll family alongside the hero and the hoagie, the chopped cheese with the heat treated as the variable, and the plain chopped cheese and its other relatives deserve their own articles rather than being crowded in here.