The egg bagel sandwich is the breakfast build stripped to its two non-negotiable parts: a hot egg and a boiled-then-baked roll. There is no required cured pork and no fixed cheese; the egg is the filling and the bagel is the structure, and everything optional is layered onto that spine rather than competing with it. That restraint is the defining decision. It makes the egg's doneness and the bagel's chew the two things the whole sandwich is judged on, because there is nothing else carrying it.
The craft is in cooking an egg to survive the bagel and in choosing the bagel to survive the egg. Scrambled, the egg is taken off soft and folded to a footprint that matches the roll so it does not slide out the back on the first bite; fried, it is cooked with the yolk set enough to handle but loose enough to be the sandwich's sauce, and a runny yolk on a dense roll is a deliberate choice that turns the bread into the thing that catches it. The bagel is split and, treated right, lightly toasted, because the boil gave it a tight chewy crumb that can take a heavy, slippery, hot filling without tearing, and a roll that was only baked goes to paste under the same load. The chew of the bagel against the give of the egg is the entire textural argument; salt, pepper, and whatever is added, cheese, a smear of cream cheese, tomato, a slice of avocado, sit on top of that without changing it. This is counter food in a city that runs on it, built in under a minute, wrapped in foil or paper, and eaten one-handed in motion, which is why the structure has to hold without a plate.
The variations are mostly a matter of what gets added to the egg-and-bagel base. A slice of cheese melted into the egg pulls it toward the classic bacon-egg-and-cheese; a cured pork added makes it that sandwich outright; the everything bagel or a different roll changes its character without changing its logic. The broader breakfast family runs from the kaiser-roll bodega build to the Southern split biscuit to the New Jersey pork roll. Each of those is a codified build with its own region defending it and deserves its own article rather than being crowded in here.