How to Make It
Heat a wood-fired pizza oven to 700° to 800° or a gas or charcoal grill to very high (about 600°).
Pulse parsley, garlic, anchovies, and 1/2 cup oil in a food processor until fairly smooth. Add remaining 2 tbsp. oil; set persillade aside.
In a medium frying pan, sauté shallots in butter until softened, 3 minutes. Add wine and cook, stirring often, until reduced to 2 to 3 tbsp., 3 to 5 minutes. Stir in lemon juice and simmer briefly, then set mixture aside.
Set mussels in a 12-in. cast-iron or steel skillet with 2 tbsp. water. Roast in pizza oven (not too close to fire, or shells will get brittle) or grill, covered, until mussels open, 3 to 4 minutes. Quickly stir in shallot mixture; cook 1 minute. Drizzle with persillade, stir, and cook (covered, if using a grill) 1 more minute.
Make ahead: Persillade, up to 1 week, chilled.
Note: Nutritional analysis is per first-course serving.