Make the pistou: To a food processor, add the garlic, basil, and ¼ cup plus 2 tablespoons of the olive oil. Process until smooth, scraping down the sides of the bowl as needed, 1 to 2 minutes.
To a large pot set over medium-high heat, add the remaining 2 tablespoons of olive oil. Add the shallots and cook until they are soft but not colored, about 2 minutes. Add the mussels, white wine and cream. Cover the pot and cook the mussels until they begin to open, 3 to 5 minutes.
Uncover the pot and add the pistou, lemon juice, liqueur and salt, then stir to combine. Divide the mussels among 6 bowls and serve with bread on the side.