The omelette sandwich is built on a filling that has been pre-set into a solid sheet, and that is what separates it from every other egg sandwich. A scrambled or bound-egg filling is loose and has to be held together by mayonnaise or butter; an omelette is already a cooked, folded, sliceable slab of egg that needs no binder at all. It goes into bread the way a slice of cheese or ham does, as a self-supporting layer cut to the shape of the loaf. The defining decision is made in the pan before the bread is anywhere near it: a thin omelette folded into a flat rectangle slips cleanly between two slices, while a thick, rounded one rolls and bulges and makes the sandwich impossible to bite without it sliding apart.
The craft is in the cook and the moisture. The omelette is taken just to set, soft inside but with no liquid egg left to weep into the crumb, then folded flat and either used warm or left to firm up cold so it slices without tearing. A wet, underdone omelette will soak the bread from the centre and a leathery overcooked one fights the soft loaf around it, so the window is narrow and it is the whole skill. The bread is soft, plain, and buttered to the edges, the butter waterproofing the crumb against any residual moisture and bridging the bland egg to the wheat. Seasoning has to go into the egg itself, salt and pepper worked through before it sets, because there is no sauce inside and a flat omelette tastes of nothing without it.
The variations are mostly things folded into the omelette before it sets rather than added to the sandwich after. Cheese melted through the egg, fried onion, ham, mushroom, or tomato each become part of the slab and travel with it. The Spanish-style version runs potato through a thick set omelette and treats a cold wedge as the filling, dense enough to need a sturdier bread. A chilli or herb omelette pushes the seasoning further. The fried-egg sandwich is the loose-yolk relative that solves a different problem entirely. Each deserves its own article rather than being crowded in here.