Egg and cheese is a study in pairing two soft savoury things and getting away with it. Bound egg is mild, fatty, and yielding; melted or thinly sliced cheese is also soft, also rich, and works in the same low, savoury register. On paper that is a mistake, two ingredients with no contrast between them, and the sandwich succeeds precisely because it does not pretend otherwise. It is deliberately a single comforting note rather than a play of opposites, and the whole craft is making that one note deep enough to carry the sandwich without a sharp counterpart to lean on.
The craft is the bind, the cheese choice, and the seasoning that does the work the missing contrast would have done. The egg is held with just enough mayonnaise or butter to cohere without slumping into a slick, since two soft fillings make a wet centre very easy to reach. The cheese is chosen to add savour and salt rather than texture: a strong Cheddar grated through the egg, or a slice melted just to the point of softening, so it distributes flavour evenly instead of sitting as a separate slab the bite has to negotiate. Because nothing in the filling is crisp, the seasoning has to be assertive, salt and pepper worked properly through the egg and the sharpness of the cheese pulling the whole thing up. The bread is plain and soft so it carries rather than competes, and butter seals the crumb against a filling that is, by design, slightly wet.
The variations stay inside the two-soft-savoury idea and mostly add a single break in the softness. A grilled version melts the cheese fully and crisps the bread for the texture the cold build refuses; chive or cress works a green note through without changing the core; a sharper or blue cheese deepens the savoury register rather than altering the structure. Those deserve their own articles rather than being crowded in here.