Egg mayonnaise is the full formal name of the made sandwich, and the formality is doing real work. This is not loose egg with a smear of something; it is a deliberately constructed filling in which hard-cooked egg has been chopped and bound with enough mayonnaise to become a cohesive, spoonable mixture, seasoned, and only then committed to bread. The completed thing is closer to a dressed salad set between slices than to egg with a condiment. That distinction is the whole identity of the form: an egg sandwich is egg that happens to have something on it, while an egg mayonnaise is a filling that was finished in a bowl and arrives as one substance.
The craft is the bind and the chop, and both are unforgiving because there is nothing else in the sandwich to carry a mistake. Chop the egg too fine and the filling turns to a pale uniform paste with no sense of the egg in it; leave it too coarse and the pieces slide apart and fall from the sides on the first bite. The useful texture keeps distinct pieces of white and yolk visible while the mayonnaise holds them as a single body. The mayonnaise is added by judgement rather than measure: too little and the mixture is dry and crumbling, too much and it slumps into a slick that soaks the crumb from the centre out. Salt and white pepper go through the mixture, not onto the bread, so the seasoning is distributed through every part of every bite. The bread is soft and plain and buttered to the edges, the butter sealing the crumb against a filling that is by design slightly wet, and the sandwich is pressed lightly so the layers settle rather than shear apart as it is eaten.
The variations are mostly the addition of one textural or sharp note the bound filling lacks on its own. Cress or chive cuts the softness; a little mustard or curry seasoning pushes the bind savoury or warm; a slice of tomato adds wet sweetness and a thinner egg-and-tomato build. Each tips the sandwich toward a named pairing with its own logic, and those deserve their own articles rather than being crowded in here.