For the filling: Heat a small skillet over medium heat. Add 1/4 cup olive oil and the prosciutto and cook, stirring often, until the prosciutto is crispy, about 3 minutes. Add the garlic and stir. Remove the skillet from the heat and cool slightly, about 5 minutes. Add the bread crumbs and toss in the oil to coat. Stir in the parsley and 1/4 cup of Parmesan.
Meanwhile, for the mussels: Heat a large Dutch oven over medium-high heat. Add the olive oil and garlic and cook, stirring often, until the garlic is lightly browned, about 2 minutes. Add the mussels and white wine. Cover and give the pot a shake. Cook, shaking occasionally, until the mussels have just opened, 2 to 3 minutes. Discard any unopened mussels.
Position a rack in the top third of the oven. Preheat the broiler to high.
Discard the top shells of the mussels and loosen the mussel meat from the bottom shell. Leave the meat in the shell and arrange on a rimmed baking sheet. Using a small spoon, pack the filling in each of the mussel shells. Drizzle with remaining 2 tablespoons olive oil and top with the remaining 1/4 cup Parmesan. Broil until the filling is golden and the cheese is melted, 1 to 2 minutes. Serve with lemon wedges if desired.