The egg salad sandwich is an egg sandwich with salad vegetables added, and the addition creates the entire problem the build exists to solve: water. Bound egg is already a slightly wet filling held in a careful balance of mayonnaise. Introduce lettuce, cucumber, tomato, or cress and you have brought a second source of moisture into a sandwich that was already moisture-managed, and the two now fight each other and the bread. The defining fact of the form is therefore not the egg and not the salad but the relationship between them. Get the water wrong and you have a cold wet slump; get it right and you have the lightest of the egg sandwiches, the bound richness cut by something fresh and crisp.
The craft is keeping the two wet things apart for as long as possible. The egg is bound a touch tighter than it would be on its own, slightly less mayonnaise, so it can absorb a little of what the vegetables will inevitably give up without going to slick. Cucumber is the worst offender and is salted and drained or sliced thin and patted dry before it goes anywhere near the filling; tomato is deseeded or kept out of a packed build entirely; lettuce is shredded and dried so it adds crunch rather than a damp layer. Sequencing matters as much as preparation: the salad sits between the bound egg and a buttered face rather than loose against the bare crumb, so the butter waterproofs the bread and the egg holds its line. It is a sandwich built to be eaten reasonably soon after it is made, because time is the one variable no amount of draining defeats.
The variations are mostly which vegetable does the cutting and how far the bind is pushed to compensate. Egg and cress keeps it minimal and peppery; egg and tomato runs wetter and sweeter and needs the tightest bind of all; a fuller salad build with lettuce and cucumber leans closest to a dressed plate between slices. Each tips the sandwich toward a named pairing with its own logic, and those deserve their own articles rather than being crowded in here.