Egg sandwich, read at its plainest, is the version with no mayonnaise at all: hard-cooked or fried egg on buttered bread, salt, pepper, and nothing further. This is the literal reading the name allows and the one most people first make, and it is a genuinely different object from the bound egg mayonnaise it is so often assumed to be. There is no dressing doing the lubrication and no bind holding the filling as one body. The butter is the only fat in the build and the egg is the only flavour, which means the defining quality of this sandwich is exposure: with two ingredients and seasoning, there is nowhere at all to put a mistake.
The craft is the egg itself and the butter under it, because that is the entire sandwich. A hard-cooked egg is sliced rather than chopped, since there is no mayonnaise to hold chopped pieces together and a sliced egg laid flat stays where it is put. The yolk wants to be cooked just to the point of setting; taken further it turns chalky and dry against bread that has no dressing to rescue it, and the grey-rimmed overcooked yolk is the single most common failure of the form. A fried egg is the other honest route, the soft yolk acting as its own sauce, which turns the sandwich into a knife-and-fork affair that floods the hand and is eaten the moment it is built. Either way the butter is structural, not garnish: spread to the edges it is the only thing lubricating a dry filling against a dry crumb and the only carrier moving the seasoning across the slice. Salt and pepper go directly onto the egg, generously, because there is no other source of savour anywhere in the build.
The variations are the act of adding back one of the things this bare version refuses. Mayonnaise turns it into the bound egg mayonnaise; cress or chive adds the crunch it lacks; a fried egg with brown sauce in a roll is the breakfast reading entirely. Each tips the sandwich toward a named pairing with its own logic, and those deserve their own articles rather than being crowded in here.