The smoked haddock sandwich is built on a fish that has to be cooked before it can go anywhere near bread, and that is the fact that defines it. Smoked haddock, the undyed kind in particular, is a firm white fillet cured over smoke but still raw, so unlike smoked salmon or smoked mackerel it cannot simply be sliced and laid down. It is poached, gently, in milk or barely simmering water until it just sets and flakes, then cooled. What ends up between the bread is therefore warm-cooked and then cold, a mild, savoury, lightly smoked flake rather than a cured slice, and the whole build is arranged around handling that flake without breaking it down to paste.
The craft is the poach and the bind. The fillet is cooked just to the point of setting and lifted before it tightens, because overcooked haddock goes dry and ropey and loses the soft layered texture that is the entire appeal. It is flaked off the skin in pieces, not mashed, and held with only as much butter or mayonnaise as makes it cohere, since over-mixing destroys the flake the poaching was meant to preserve. The smoke is gentle, so the counter is measured rather than aggressive: a little lemon, a turn of pepper, sometimes a leaf of watercress for a green bite against an all-soft filling. The bread is soft and plain, white or brown, buttered to the edges so the crumb is sealed against a faintly moist filling, because a delicate flake wants a yielding carrier and a heavy crust would only bully it.
The variations stay close to the cooked-and-flaked idea and change the binding or the smoke around it. A version bound with chopped egg moves it toward a kedgeree register, warm spice and rice traded for bread. A thin smear of crème fraîche in place of mayonnaise keeps it lighter and lets the smoke read more clearly. The dyed fillet, brighter and more strongly cured, pushes the smoke forward at the cost of the subtlety the undyed version trades on. Each deserves its own article rather than being crowded in here.