How to Make It
Melt 1/2 cup of the butter in a large pot over medium. Add 1/2 cup of the shallots and 1/2 tablespoon of the garlic. Cook, stirring occasionally, until translucent, about 5 minutes. Add mussels, wine, 2 of the bay leaves, and 1 of the thyme sprigs; cover and bring to a simmer. Cook until mussels open, about 8 minutes. Pour through a colander into a large bowl, reserving solids and cooking liquid. Discard any unopened mussels. Cool mussels about 10 minutes.
Cook leek, celery, onion, carrots, 1/2 teaspoon of the salt, and remaining 2 bay leaves, 1 thyme sprig, 1/2 cup butter, 3/4 cup shallots, and 1/2 tablespoon garlic in a large pot over medium, stirring occasionally until tender, about 12 minutes. Add reserved cooking liquid; increase heat to medium-high, and cook until reduced to about 8 cups, about 15 minutes. Stir in half-and-half, and bring to a simmer.
Whisk egg yolks in a medium bowl. Slowly add 1 cup of warm soup to yolks, whisking constantly. Slowly stream yolk mixture into soup, stirring with a whisk. Stir in Espelette, cayenne, and remaining 2 teaspoons salt.
Divide mussels among 12 bowls, and ladle hot soup into each.