Preheat the oven to 350°. Cut 1 of the baguette halves crosswise into 1/4-inch-thick slices. Spread the slices on a rimmed baking sheet and bake for 15 minutes, or until golden. Remove the crust from the remaining baguette half. Cut the crust into 1-inch pieces and reserve 1 cup. Cut the crustless baguette into 1-inch dice; spread the bread cubes on another baking sheet and bake for 10 minutes, or until crisp; reserve.
In a mini food processor, combine the 1 cup of reserved crust pieces with 1/4 cup of water and let soak until moistened. Add the piquillo peppers and half of the garlic and process until smooth. With the machine on, add the olive oil in a slow steady stream and puree until thick and creamy: the rouille should be the consistency of mayonnaise. Add the lemon juice and season with salt and cayenne. Thin the rouille with 2 tablespoons of hot water if it seems too thick.
Melt 2 tablespoons of the butter in a large saucepan. Add the remaining half of the garlic and cook over moderately high heat until golden. Add the mussels, wine and saffron, cover and cook until the mussels begin to open, about 6 minutes; transfer to a bowl as they open and let cool slightly. Remove the mussels from their shells and place in a small bowl; discard the shells and any mussels that do not open. Strain the cooking liquid into a heatproof measuring cup; discard the solids. Add water, if needed, to make 2 cups of liquid.
Wipe out the saucepan and melt the remaining 2 tablespoons of butter in it. Add the onion and celery and cook over moderate heat until softened, about 6 minutes. Stir in the tomato paste and cook until just glossy. Add the diluted clam juice and the reserved cooking liquid, leaving behind any grit. Bring to a boil and simmer for 20 minutes.
Add the cream and the reserved toasted bread cubes to the broth and simmer until the bread is softened, about 5 minutes. In a blender, puree the soup in batches until very smooth. Return the soup to the saucepan, season with salt and cayenne and simmer until reduced to 8 cups, about 10 minutes.
Stir the mussels into the soup and bring to a simmer. Ladle into shallow bowls. Spread the toasts with the rouille and serve.
Make Ahead: The rouille, mussels and soup can be refrigerated separately overnight. Store the toasts in an airtight container. Recrisp in a 325° oven if necessary.
Notes: Jarred piquillo peppers, from the Navarre region of Spain, are available at specialty food shops. Piquillos are richer in flavor than roasted red bell peppers and have a bit of heat. The best substitute is jarred roasted red peppers plus 1/4 teaspoon of hot paprika.