MELT 4 TABLESPOONS OF BUTTER in a large, heavy skillet over medium heat. Add the fennel, cover and slowly braise, stirring occasionally, until the fennel is completely softened and lightly browned, about 30 minutes. Transfer it to a food processor fitted with a steel blade and puree. Set aside. Melt the remaining butter in a heavy large saucepan over medium-low heat. Add the flour and stir with a wooden spoon until the flour is cooked but still light in color, 3-to-4 minutes. Slowly add the stock to the flour mixture, stirring constantly. Bring it to a boil, then reduce the heat to low and simmer gently for 30 minutes. Stir in the fennel, then pass the soup through a fine strainer, if desired.
(The soup may be prepared ahead of time, to this point.)
Just before serving, bring the soup to a simmer. Combine the egg yolks and cream in a small bowl, and beat well. Gradually add about 2 cups of soup to the yolk mixture. Slowly blend the yolk mixture back into the remaining soup. Be careful not to let the soup boil or the eggs will curdle. Season to taste with salt and pepper. Ladle into warm bowls and garnish with chives.