Step 1Place the mussels, wine and 1 cup cold water in a large saucepan over moderately high heat. Cover and cook until the shells open, 4 to 6 minutes. Strain the mussels into a colander, collecting the juices in a bowl placed below.
Step 2Heat the olive oil in a saucepan over low heat, add the garlic, and cook, stirring constantly, until pale gold, 3 to 4 minutes (do not let brown).
Step 3Add the mussel juice to the garlic, raise the heat to medium and bring to a boil. Cover, lower the heat to very low, and simmer for 10 minutes.
Step 4Meanwhile, remove the mussels from the shells. Lightly toast the bread.
Step 5Remove the soup from the heat. Combine the egg yolk, vinegar and a couple of tablespoons of the soup in a mixing bowl and beat vigorously with a whisk until the mixture gets foamy. Slowly pour the mixture back into the remaining soup, continuing to beat with a whisk.
Step 6To serve, place a few baguette slices, 3 to 4 tablespoons of grated cheese and some mussels on the bottom of 4 wide soup bowls, cover with soup, and dust with piment d'Espelette.