In 3-quart saucepan, heat milk with bay leaf over medium heat until nearly simmering. Add fish and cook until just starting to flake. Using a slotted spoon, remove fish to paper towels to drain; set aside. Discard bay leaf. Transfer milk to heatproof mixing bowl or other container; set aside.
Wash saucepan and return to medium heat. Add butter and cook until melted. Add onion and cook, stirring frequently, until nicely softened but not browned, about 6 minutes.
Return milk to saucepan and add potatoes. Return milk to bare simmer. Then cook until potatoes are tender, about 10 minutes; stir and scrape saucepan occasionally to prevent milk from scorching.
Using a potato masher or large wooden spoon, crush potatoes in pot until partially mashed. You want a good range of potato bits, some so small they thicken the broth but also larger chunks.
Remove and discard any skin or bones from fish; then flake into the soup. Add cream. Season with salt and pepper, and, if desired, stir in pinch of cayenne. Serve, sprinkling chives on top of each bowl.