Add potato quarters to a medium pot with 4 cups water and salt. Boil until soft but not falling apart when pierced with a knife, about 7-8 minutes. Strain potatoes (or use a slotted spoon to pull them), reserving water.
Cut fish fillet into even sized pieces and add to potato water with bay leaves. Bring to a simmer and cook until soft and flaky, about 8-10 minutes. Remove fish and set aside. Pour fish/potato water into bowl and set aside.
Add butter to original pot and melt over medium heat. Add onion, leek, garlic, thyme, salt, and pepper and sauté until soft (not brown). Add flour and mix to combine, cooking for just a minute. Add potato/fish water, plus 2 cups milk and stir until well combined. Let simmer for 15 minutes.
While soup is simmering, flake fish. Add fish and potatoes back to broth and simmer 5 minutes longer. Salt and pepper to taste and top with fresh herbs.