The Cullen skink sandwich is a soup made solid enough to put between bread, and the reduction is the entire trick. Cullen skink is a thick Moray soup of smoked haddock, potato, onion, and milk or cream, and a soup cannot be a sandwich filling. So it is cooked down hard, the liquid driven off and the potato mashed through it, until what was a bowl becomes a dense, smoky, spreadable mass that holds its shape against a slice. The smoked haddock is doing the flavour and the potato is doing the body, and turning one into the other without losing the smoke is the whole point of the build.
The craft is in the reduction and the seal. The filling has to be taken further than a spoonable soup and cooled, because warm and loose it soaks straight through any bread and the sandwich fails before it is cut; reduced thick and chilled, it behaves like a coarse fish paté and can be spread or piled. The smoked haddock is flaked through rather than blended to a smooth purée, so the bite keeps the muscle and the smoke reads as fish, not as a flavouring. Butter on the bread is structural here, a waterproof layer between a crumb and a filling that is still, by nature, damp, and the bread is plain and sturdy so it carries a heavy, savoury load without competing with the smoke. A squeeze of lemon or a little chopped chive cuts the richness so the sandwich does not read as one heavy creamy note.
The variations stay close to the soup it comes from. A version that keeps the filling looser and serves it warm on toast, closer to the bowl than the sandwich; a flaked smoked-haddock build without the potato body, lighter and more directly fish; the Finnan haddie and Arbroath smokie sandwiches that use a differently cured smoked haddock as a plain statement rather than a reduction. Each is its own Scottish fish sandwich and deserves its own article rather than being crowded in here.