On an American brunch menu the croque monsieur stops being a sandwich you pick up and becomes a plate you cut into, and that shift is the whole American reading of it. The French original is a quick counter food eaten with a knife and fork almost by accident. The bistro version makes the knife and fork the point: more béchamel, a heavier cap of grated Gruyère, a longer time under the broiler, and a finish that arrives lacquered and bubbling on a warm plate rather than wrapped in paper. The sauce, not the ham, is what the kitchen leans on. Béchamel goes on the bread, soaks inward, and binds the structure into one mass, and the American version simply uses more of it so the sandwich reads as rich and composed instead of fast and plain.
It works because every component is sauced or melted into continuity rather than stacked. Thin-sliced ham and a firm, sliceable cheese scale cleanly to the bread, but on their own they would be an ordinary ham and cheese. The béchamel is the difference: it coats the interior, the bread drinks it, and the cheese on top blisters into a brown crust under the broiler while the center stays molten. The bread has to be sturdy enough to carry a wet sauce without going to paste, which is why a close-crumbed pain de mie or a dense white loaf is the right base and an airy roll is not. A bistro can build a tray of these, hold them, and run them under the salamander to order, which is the entire reason the format survives a Sunday rush.
The variations are codified by a single swap each. Crown it with a fried or poached egg and it becomes the croque-madame, the egg standing in for the hat. Push the same logic toward the griddle and you have the patty melt and the tuna melt, close American relatives that solve the same melt-and-bind problem at lower heat. The Monte Cristo batters and fries a ham-and-cheese version into something between a sandwich and French toast. Each of those is one decision away from this one, and each deserves its own article rather than being crowded in here.