In a large, heavy-bottomed saucepan, heat the oil over high heat until smoking. Add all the chicken parts and brown all over, stirring to avoid burning. Remove the chicken and reserve. Add the carrots, onions, and celery to the pot and cook until soft and browned. Return the chicken to the pot and add 3 quarts of water, the tomato paste, peppercorns, and parsley. Stir with a wooden spoon to dislodge the browned chicken and vegetables bits from the bottom of the pan. Bring almost to a boil, then reduce heat and cook at a low simmer until reduced by half, about 2 hours, occasionally skimming excess fat. Remove from heat, strain, and press on the solids with the bottom of a ladle to extract out all liquids. Stir the stock to facilitate cooling and set aside. Refrigerate stock in small containers for up to a week or freeze for up to a month.
Yield: 1 1/2 quarts
Roll the pasta out to the #4 setting, and form into gramigne. To form into gramigne, cut the dough into fingers that are 1/2-inch thick and 3 inches long. Starting with the palm of your hand, roll the dough pieces lengthwise. It should look like a piece of squiggly spaghetti.
In a 12 to 14-inch saute pan, heat the olive oil and butter together over high heat. Add the onion and cook until soft and golden brown, about 3 minutes. Add the sausage and cook over high heat, draining the excess fat if necessary, until the sausage is very browned and somewhat crispy, about 10 minutes.
Add the wine and, stirring frequently, allow it to evaporate completely. Stir in the milk, chicken stock, tomato paste and salt and pepper, to taste, and reduce heat to a simmer. Cook for 15 minutes.
Meanwhile, bring 6 quarts of water to a boil and add 2 tablespoons salt. Drop the gramigne into the boiling water and cook for 3 minutes. Drain the pasta, reserving the cooking water, and add the pasta to the sausage mixture. Toss over high heat 1 minute to coat, adding a splash of the reserved pasta water if necessary to keep the condiment from getting too "tight". Divide equally among 6 warmed pasta bowls, top with grated Parmigiano-Reggiano, and serve immediately.
Mound 3 1/2 cups 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 the olive oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour, starting with the inner rim of the well. As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. 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 and discard any leftover bits. Lightly reflour the board and continue kneading for six more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature. Roll or shape as desired.