To prepare ragu: Wash duck legs and remove all fat. Pat dry.
In a thick bottomed casserole or Dutch oven, heat olive oil until smoking. Add duck legs and cook until brown on all sides and remove, about 10 to 12 minutes. Add onion, carrot, garlic and celery and cook until softened, about 7 to 9 minutes. Add wine, tomatoes, chicken stock and dried mushrooms and bring to a boil. Add duck legs and return to boil, lower heat, cover and allow to simmer for 1 hour. Remove duck legs and allow to cool. Pull all meat off the bones and return to the pot without the bones. Simmer uncovered for 30 minutes, or until quite thick. Season with salt and pepper and set aside.
To prepare dish: Fill a large pot with 6 quarts water and add 2 tablespoons salt. Cook bigoli until al dente, about 8 to 9 minutes. Heat sauce, drain pasta and toss in pan. Toss to coat and serve immediately.
To make pasta: Make a mound of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs and milk mixture. Using a fork, beat together the eggs and milk mixture and begin to incorporate the flour starting with the inner rim of the well.
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.
Note: Do not skip the kneading or resting portion of this recipe. They are essential for a light pasta.
To make bigoli: Knead dough until smooth and tight and let rest 20 minutes. Cut dough into 6 and run through a meat grinder set to smallest extrusion size. As pasta exits, cut it into 12-inch pieces and immediately dust with flour. Lay out on to cookie sheet dusted with cornmeal, being careful to keep the strands separate. Finish all 6 pieces the same way.
Recipe Courtesy of Mario Batali