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In a heavy-bottomed casserole or Dutch oven, heat the olive oil until smoking. Add the duck legs and cook until brown on all sides and remove, about 10 to 12 minutes. Add the onion, carrot, garlic and celery and cook until softened, about 7 to 9 minutes. Add the wine, tomatoes, chicken stock and dried mushrooms and bring to a boil. Add the duck legs and return to a boil, lower the heat, cover and allow to simmer for 1 hour. Remove the duck legs and allow to cool. Pull all meat off the bones and return to pot, without the bones. Simmer uncovered for 30 minutes, or until quite thick. Season with salt and pepper and set aside.
As you expand the well, keep pushing the flour up to retain the well shape. Do not worry that this initial phase looks messy. The dough will come together when half of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up any leftover crusty bits. Discard these bits. Lightly flour the board and continue kneading for 3 more minutes. The dough should be elastic and a little sticky. Continue to knead for another 3 minutes, remembering to dust your board when necessary. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature.
To make the Bigoli: Knead dough until smooth and tight, and let rest 20 minutes.
Note: Do not skip the kneading or resting portion of this recipe. They are essential for a light pasta. Cut the dough into 6 pieces, and run each piece through a meat grinder set to the smallest extrusion size. As the pasta exits, cut it into 12-inch pieces and immediately dust with flour. Lay out on a cookie sheet dusted with cornmeal, being careful to keep the strands separate. Finish all 6 pieces the same way.