Collect all ingredients required for the soup.

In a large saucepan, combine the milk, parsley sprigs, bay leaf, and smoked haddock fillet. Bring to a gentle boil over medium heat. Lower the heat to a gentle simmer for about 3 minutes. Remove from heat and allow it to infuse for 5 minutes.

Finely chop the reserved parsley leaves.

Remove the smoked haddock from the milk and set aside. Strain the milk through a fine mesh strainer to remove herbs and solids.

In another large saucepan over medium-low heat, melt the unsalted butter and add the finely chopped onion. Cook until soft and translucent, about 5 minutes.

Pour the strained milk into the saucepan with the onions and butter. Stir in the mashed potatoes until fully dissolved and the soup thickens slightly.

Flake the smoked haddock into bite-sized chunks, removing any bones, and gently add to the soup.

Lower the heat to a gentle simmer. Add chopped parsley leaves and cook until the fish is heated through, about 5 minutes. Season with kosher salt and freshly ground black pepper to taste.

Ladle soup into bowls and garnish with reserved parsley leaves and additional ground black pepper. Serve warm with crusty bread if desired.