The American wit is a cheesesteak resolved down to its two most argued variables: American cheese and fried onions, ordered together in one breath. The Philadelphia counter runs on a grammar where the cheese is a choice and the onions are a yes-or-no, and this build settles both at once. Wit means with onions; the American specifies the cheese against the provolone and the processed cheese sauce that are its rivals. What defines the sandwich is that decision and what it does on the griddle: American cheese melts into the chopped beef rather than sitting on it, and the onions are fried into the same surface, so the three components fuse before the roll ever closes.
The craft is decided on the flat-top in about ninety seconds. Thinly sliced beef is chopped and seared fast and hot so it stays tender rather than going to leather; sliced too thick or worked too long it turns to rope. The American slice is laid on while the meat is still on the steel, because its formulation flows instead of splitting, so it coats the beef and binds the pile rather than draping over it as a sheet. The onions are not a topping added at the end; they are griddled down until soft and sweet and folded through the meat so they are inside the mass, not on it. The roll is a structural component: a long Philadelphia roll with a tender interior and a crust that has enough spine to hold a heavy, greasy, molten filling without tearing or dissolving into the grease. The whole thing is assembled hot and eaten immediately, because the cheese and onions are doing the binding and a cooled wit slumps.
The variations are the rest of the ordering grammar. Witout drops the onions and leaves the cheese to do the work alone. Whiz wit swaps the American for processed cheese sauce and keeps the onions. Provolone wit trades melt for a sharper, firmer cheese that behaves differently against the beef. The chicken cheesesteak runs the same method on a different protein. Each of those is its own order with its own logic and deserves its own article rather than being crowded in here.