The scrambled egg sandwich is defined by the texture of a filling that does not want to stay where it is put. Eggs are beaten and cooked slow and soft so they set in loose, glossy curds rather than a firm sheet, then piled warm into bread. That softness is the whole appeal and the whole problem. A loose scramble is rich, tender, and faintly creamy in a way a hard-cooked egg never is, but it carries free moisture, and free moisture against bread is a clock. The defining fact of this sandwich is the race between the pleasure of a soft scramble and the wet base it produces, and most of the craft is about losing that race as slowly as possible.
The craft is the cook and the seal. The eggs are taken off the heat while still slightly underdone because they carry on setting in their own warmth on the way to the bread, and a scramble cooked dry to be safe is rubbery and weeping at once. They are seasoned in the pan and kept loose rather than stiff. The bread is where the sandwich is won: buttered firmly to the edges so the fat waterproofs the crumb and buys time before the egg soaks through, and chosen soft so it matches a filling with no chew of its own. Toast holds longer than fresh bread because its drier, firmer face resists the moisture, which is why a scrambled egg sandwich is so often made on toast rather than a soft slice. The eggs go in as a thick, even layer and the sandwich is eaten promptly, because this is a filling that does not improve with waiting.
The variations stay close to the egg. A crunch worked in, chive or cress, breaks an all-soft filling that on soft bread reads as a single note. A grind of pepper or a little smoked fish lifts it. The wider egg sandwiches, hard egg bound with mayonnaise, an omelette folded cold, the Scotch egg sliced between slices, are the same protein met at a firmer texture that does not run. Each of those deserves its own article rather than being crowded in here.