Trim bases from fennel bulbs; reserve fronds for garnish. Cut each bulb in half through base. Cut out core from each half. Slice fennel lengthwise into 1/4" slices; cut slices in half crosswise.
Combine fennel and half-and-half in a medium saucepan. Bring to a boil; reduce heat, and simmer 25 to 30 minutes or until fennel is very soft. Drain fennel, reserving cooking liquid. Place fennel and 2 tablespoons cooking liquid in a blender or food processor; process 30 to 45 seconds or until smooth.
Meanwhile, bring potato and water to cover to a boil in a large saucepan; cook 15 to 20 minutes or until tender. Drain potatoes, and return to saucepan. Add butter, fennel puree, salt, and pepper; mash well with a potato masher or hand mixer until smooth. For creamier potatoes, add reserved cooking liquid, 1 tablespoon at a time, to desired texture. Garnish with chopped fennel fronds, if desired.