Hard-boiled egg carries this one, and the form it takes depends on whether that egg is sliced or mashed. In the sliced build, rounds of hard-boiled œuf are laid in a layer on a buttered loaf with salt and pepper; in the bound build, the egg is chopped and folded with mayonnaise into a soft, spreadable mass. Either way the egg is the whole sandwich, and what lifts it past a plain filler is how its richness is handled, not what is added on top of it.
The logic follows from the egg itself. A hard-boiled egg is firm protein with a dry, crumbly yolk, so it brings savor and body but a texture that goes chalky on its own. That is why fat is the structural decision. In the sliced version, butter underneath bridges the dry yolk to the bread and keeps the rounds from sliding; in the bound version, the mayonnaise is the binder and the moisture at once, turning a crumbly thing into a paste that grips the crumb. The constraint in both is seasoning and bread. Egg is bland until salted properly, and the loaf needs enough crust to hold a soft, heavy filling that offers no structure of its own. The bound version travels better and softens the bread slower; the sliced version stays cleaner and crisper but wants to be eaten sooner, before the rounds shift and the loaf gives.
Variations stay close to the egg. A version with a few sprigs of chive or cress adds a sharp green note against the richness; one with a leaf of lettuce and a slice of tomato pushes it toward a fuller cold sandwich; the plainest is sliced egg, butter, salt, and bread alone. Each holds the hard-boiled egg as the fixed point and changes only what cuts or carries it. The Sandwich aux Œufs belongs with the plant-forward and meatless builds the catalog groups under Sandwich Végétarien, the tradition that treats a single non-meat component as the lead. Its specific contribution is a sandwich that lives entirely on how the egg's dry richness is bound and seasoned, with nothing else to lean on.