Pappardelle with Duck Ragù

Name: Pappardelle with Duck Ragù
Ingredients: 2 tablespoons extra-virgin olive oil; Three 1-pound duck legs, skin and excess fat removed; Kosher salt; Freshly ground pepper; 6 large shallots, finely chopped; 1 large carrot, chopped; 2 celery stalks, chopped; 3 ounces prosciutto, finely chopped; 2 rosemary sprigs; 2 tablespoons thyme leaves; 4 garlic cloves, very finely chopped; 1/4 ounce dried porcini mushrooms, ground to a powder in a spice grinder; 1 1/2 cups dry white wine; 3 cups chicken stock; 8 ounces dried pappardelle noodles; 1/2 cup fresh ricotta cheese; 2 tablespoons freshly grated pecorino cheese, plus more for serving