Prep the Fennel: Trim the long stalks from the fennel bulbs. Finely chop a few of the feathery fronds and set them aside for garnish.
Cut the tough root end off each bulb, then quarter them and slice the quarters into 1/2-inch thick pieces. You should have about 8 to 9 cups (2160 ml) of sliced fennel. 2. Build the Flavor Base: In a large stockpot or Dutch oven, melt 3 tablespoons (45 ml / 1.5 fl oz) of the butter over medium heat.
Add the diced onion and sauté for 5 minutes until softened and translucent.
Add the sliced fennel and diced potato, stirring often, and continue to sauté for another 5 minutes. 3.
Simmer to Softness: Pour in the vegetable stock, salt, and a generous amount of freshly ground pepper. Increase the heat to bring the soup to a boil.
Once boiling, reduce the heat to low, cover the pot, and let it simmer for 20 to 30 minutes, or until the fennel and potato are very tender when pierced with a fork. 4. Create the Silky Texture: Carefully transfer the soup to a blender or use an immersion blender to purée until completely smooth. For an extra-velvety finish, pour the puréed soup through a medium-mesh sieve back into a clean saucepan, pressing on the solids to extract all the liquid. 5.
Finish and Serve: Gently heat the strained soup over low heat.
Stir in the remaining 2 tablespoons (30 ml / 1 fl oz) of butter until melted, then stir in the milk.
Remove from the heat and whisk in the sour cream until fully blended and the soup is creamy.
Taste and adjust the salt and pepper as needed.
Serve immediately, garnished with the reserved fennel fronds.
Don't discard your leftover fennel fronds! Store them in a plastic bag in the refrigerator and chop them into salads or use them as a garnish for other dishes for a fresh, anise-like flavor.