How to Make It
In a mini food processor, combine the basil, walnuts and whole garlic clove and pulse until fine. Add the Parmesan and 2 tablespoons of the oil and process until smooth. Transfer the pesto to a medium bowl and add the butter. Stir in the tomatoes and season with salt and pepper.
In a very large, deep skillet or large soup pot, heat the remaining 1/4 cup of olive oil until shimmering. Add the shallots and sliced garlic and cook over high heat, stirring, until lightly golden, about 4 minutes. Add the white wine and a generous pinch each of salt and pepper and bring to a boil. Add the mussels and stir for 1 minute. Cover and cook, stirring occasionally, until all of the mussels have opened, about 6 minutes. Add the pesto and tomatoes and stir until the butter is thoroughly melted and the mussels are evenly coated. Transfer the mussels and broth to bowls and serve with crusty bread.