Stir together flour and salt and add olive oil, stirring with a whisk. Add eggs and mix with hands until sticky. Add water 1 tablespoon at a time until dough stays together. Knead 5 to 8 minutes until pliable. Cover and allow to rest 15 minutes.
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 pot, without the bones. Simmer uncovered for 30 minutes, or until quite thick. Season with salt and pepper and set aside.
In a large pot, bring 6 quarts of water to boil and add 2 tablespoons salt. Roll the pasta dough out to its thinnest setting and then cut by hand into pappardelle, about 1 to 11/4 inch thick.
Heat 2 cups duck ragu in a 12 to 14 inch saute pan until quite hot. Boil pappardelle until cooked, about 1 minute and drain well. Put hot pappardelle into pan with duck ragu and toss well. Pour into serving bowl and serve immediately.
Recipe courtesy of Mario Batali