How to Make It
Preheat the oven to 350°. In a large enameled cast-iron casserole, heat the olive oil until shimmering. Season the duck legs with salt and pepper and add to the casserole. Cook over moderately high heat, turning once, until browned, about 8 minutes; transfer to a plate.
Pour off all but 2 tablespoons of fat from the casserole. Add the shallots, carrot, celery, prosciutto, rosemary, thyme and garlic. Cook over moderately high heat, stirring occasionally, until the vegetables are softened and just starting to brown, about 5 minutes. Stir in the porcini powder and wine. Simmer, scraping up any browned bits from the bottom of the casserole, until slightly reduced, about 3 minutes. Add the chicken stock and bring to a boil. Return the duck legs to the casserole, cover and braise in the oven until the legs are very tender, about 1 hour and 30 minutes; turn the duck legs halfway through braising.
Transfer the duck legs to a plate and let them cool slightly. Remove the duck meat from the bones and shred into bite-size pieces. Return the duck to the ragù and season with salt and pepper; keep warm. Discard the rosemary stems.
In a large saucepan of salted boiling water, cook the pappardelle until al dente. Drain, reserving 2 cups of the cooking water. In a bowl, whisk the ricotta with 2 tablespoons of the cooking water until smooth.
Add the pasta to the ragù along with 3/4 cup of the reserved cooking water and the 2 tablespoons of pecorino. Cook over moderate heat, stirring gently, until the pasta is hot and coated with sauce; add more cooking water if the sauce is too thick. Transfer the pasta to shallow bowls, top with the ricotta and serve, passing additional pecorino at the table.