The sausage, egg, and cheese is the bodega breakfast built around a flat sausage patty instead of bacon, and that one swap changes the whole structure. Where bacon is a thin, brittle layer that disappears into a soft sandwich, a breakfast sausage patty is a substantial slab with its own height and bite, so it becomes the spine of the build rather than an accent. It is cooked on a flat-top griddle that never cools, the egg is broken and folded next to it to roughly the footprint of the roll, and a slice of American cheese is laid on the hot egg so it melts down and glues the patty, the egg, and the bread into one mass that survives being wrapped in foil and eaten with one hand.
The craft is in the griddle and the carrier. The sausage patty is pressed thin and wide so it matches the roll and cooks through fast, its rendered fat seasoning the griddle and, by extension, the egg cooked beside it. The egg is kept soft and folded rather than fried hard, because a rubbery egg breaks the cohesion the cheese is trying to build. The cheese is melted onto the egg on the griddle, not laid on cold at the end, which is the small move that turns three components into a single structural object. The roll is a soft-crumbed kaiser or a hero, split and often griddled on the cut faces so it has a little structure against a wet, fatty filling, then dressed with salt, pepper, and frequently a squirt of ketchup or hot sauce before the foil seals and steams it on the walk.
The variations are mostly the carrier and the heat. The roll becomes a bagel when a chewier base is wanted, or a hard roll, or a wrap; a croissant version trades the savory roll for a buttery, flaky one and tilts the sandwich richer. The Southern reading moves the same fillings onto a split biscuit, and the sweet-sausage and hot-sausage builds change the patty's seasoning rather than the frame. Each of those deserves its own article rather than being crowded in here.