In a large saucepan, bring 8 cups water to a boil over medium-high heat. Add 1 teaspoon salt and potatoes. Cook until tender, 8 to 10 minutes. Drain potatoes, discarding liquid.
In a large Dutch oven, melt 2 tablespoons butter over medium heat. Add haddock; cook for 1 minute on each side. Remove haddock. Use a fork to flake one-fourth of haddock.
Melt remaining 2 tablespoons butter in Dutch oven. Add garlic, onion, leeks, bay leaves, parsley, thyme, pepper, mustard, and remaining 1. teaspoons salt; cook
until fragrant, about 2 minutes. Add cream. Bring to a boil over medium-high heat; reduce heat and stir in potatoes. Use a potato masher to mash potatoes. Add three-fourths haddock, and simmer soup, stirring occasionally, for 15 minutes.
To serve, ladle into bowls. Top with flaked haddock, and garnish with chopped parsley and thyme, if desired.
Notes
Note: If smoked haddock is not available, any smoked white fish may be substituted.
Recipe by Victoria at https://victoriamag.com/cullen-skink-recipe/