Notes: Chop feathery green fennel leaves and sprinkle onto servings.
Rinse fennel. Trim and discard stems and bruised portions (save leaves; see notes). Chop fennel heads.
In a 4- to 5-quart pan over medium-high heat, frequently stir fennel, onions, and butter until vegetables are golden, about 25 minutes.
Add potatoes and broth. Cover and bring to a boil over high heat, then reduce heat and simmer, stirring occasionally, until potatoes mash easily when pressed, about 20 minutes.
In a blender, whirl soup mixture, a portion at a time, until smoothly puréed. Return to pan. Add half-and-half, and liqueur to taste. Stir over medium-high heat until hot, about 3 minutes.
Ladle into bowls and season to taste with salt and pepper.
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