STEP 1
In a large pot, warm the duck fat over medium heat. Add the shallots and garlic and cook, stirring occasionally, until softened, but not browned, about 5 minutes. Turn the heat to medium high, add the mussels and wine, cover, and cook, shaking the pan occasionally, until the mussels open, 3–5 minutes. Discard any that do not open.
STEP 2
Using tongs or a slotted spoon, transfer the mussels to a cutting board. Pour the cooking liquid through a fine sieve into a small pot and scrape the shallots and garlic into a medium bowl. Bring the cooking liquid to a boil and cook until reduced to 3⁄4 cup, about 15 minutes. Remove the pan from the heat and let the reduction cool completely.
STEP 3
Pour 1⁄4 cup of the reduction into the reserved shallots and garlic; stir in the olive oil, chives, parsley, tarragon, lemon rind, and lime juice. Season to taste with salt and piment d’Espelette.
STEP 4
Remove and discard the empty top shells from each mussel and transfer the bottoms to a serving platter. Spoon the vinaigrette over the mussels and serve immediately.