Trim and discard root ends and coarse tops from leeks. Cut leeks in half lengthwise and rinse under running water, flipping layers to flush out grit; drain, then thinly slice crosswise.
Rinse fennel. Trim off and discard root ends and stalks; reserve 2 or 3 sprigs of feathery green tops for garnish. Chop fennel heads.
In a 6- to 8-quart pan over medium heat, melt butter. Add leeks and chopped fennel heads, cover, and stir occasionally until vegetables are very tender, 10 to 12 minutes.
Add broth and bay leaf to pan. Bring to a boil over high heat. Scrub potatoes and cut into 1/2- to 3/4-inch cubes.
Add potatoes to broth mixture and return to a simmer; reduce heat, cover, and simmer, stirring occasionally, until potatoes are tender when pierced, 15 to 20 minutes.
In a bowl, whisk milk, flour, 1/2 teaspoon salt, and 1/8 teaspoon pepper until smooth. Add to potato-broth mixture and stir over high heat until boiling, about 5 minutes. Add more salt and pepper to taste.
Serve in deep bowls. Garnish with reserved fennel sprigs and chives.