The Finnan haddie sandwich is defined by a step that happens before any bread is involved: the fish has to be cooked. Finnan haddie is haddock cold-smoked over peat and hardwood, which cures and perfumes the flesh without garlanding it through, so it comes off the smoke pale, firm, and essentially raw. This is the fact that separates it from every hot-smoked fish that goes straight into a sandwich. A Finnan haddie cannot simply be flaked between slices; it must first be gently poached, traditionally in milk, until it turns opaque and yields. The sandwich is the back half of a small cooking process, and its character is set in the pan, not at the board.
The craft is the poach and the restraint that follows it. The fish is cooked slowly in milk or water barely moving, because hard heat turns a cold-smoked haddock tight and dry and forfeits the soft, clean, faintly peat-scented flesh that is the whole reason to use it. Once poached it is drained well, skinned and boned, and lifted into large flakes kept loose so they stay moist rather than packing into a dense block between the bread. From there the discipline is the same one every good smoked-fish sandwich obeys: the haddock carries salt and a gentle smoke on its own, so the butter is there to lubricate and bridge rather than to flavour, and the bread is soft and plain so it stays behind the fish. A little of the warm poaching milk worked back through the flakes, or a thin butter sauce, keeps a lean fish from reading dry between slices.
The variations stay tied to the poached fish. A soft-poached egg turned through the flakes pushes it toward the kedgeree register on bread; a horseradish or mustard cream cuts the smoke for those who want it sharpened; the hot-smoked Arbroath smokie is the louder, fully cooked relative of the same coast. Those are distinct enough to deserve their own articles rather than being crowded in here.