Egg and anchovy is defined by a single deliberate collision: an intensely salty cured fish driven into a filling whose whole character is softness. Hard-cooked egg, bound just enough to cohere, is mild, fatty, and almost flat on its own, and the anchovy, whether mashed through as fillets or carried as a scrape of the concentrated relish made from them, supplies everything the egg lacks. It is salt, it is umami, it is depth, applied in a quantity small enough to season rather than dominate. The sandwich exists because plain egg mayonnaise is gentle to the point of being dull, and the anchovy is the one sharp savoury note that makes it interesting without giving it any new texture.
The craft is the ratio and the seasoning, and it is unforgiving because there is so little else in the sandwich to hide behind. The anchovy is strong enough that a fraction too much turns the whole filling fishy and harsh, and a fraction too little leaves it back where it started, so it is worked through the egg in measured amounts and tasted as it goes. Because anchovy already carries the salt, the egg is otherwise barely seasoned, and the bind stays light so the filling holds without slumping into a slick. The bread is plain and soft, since both components are soft and an assertive crust would only argue with a filling that has no crunch of its own, and butter seals the crumb against a filling that is, by design, a little wet.
The variations stay close to the egg and the salt-fish idea. The Gentleman's-relish version uses the spiced anchovy paste in the smallest possible scrape for concentrated depth; a build with a few whole fillets laid on top reads sharper and saltier in bursts; a cress or chive addition breaks the all-soft texture without touching the savoury core. Those deserve their own articles rather than being crowded in here.