The fish used is a smoked haddock called a “Finnan Haddock.”
You put the fish in a pot, with just enough boiling unsalted water to cover it. Leave the skin on the fish. Bring the water back to a boil, add chopped onion, and simmer for a few minutes to cook the fish. Take the fish out, remove the skin, flake it, and debone. Put the bones back in the pot of water, and simmer for an hour to make a stock. Then strain, discarding the bones, return the stock to the pot, heat again, add heated milk, and the fish. Let simmer for a few minutes, then thicken with leftover mashed potato, and adjust the taste with butter, salt and pepper. The soup can be served garnished with parsley.
Some versions have you skip the step of making a stronger stock with the bones.
Cullen Skink is often used as a starter at more formal dinners in Scotland.
You can use any smoked haddock. Finnan Haddock is cold-smoked, so it’s not cooked.
“Skink” originally meant a soup made from beef shank. Skink is the Scots version of the English word “shank.”