Step 1Place the large pieces of carrot and celery and the carrot, celery and onion trimmings in a medium saucepan with 1 bay leaf and the peppercorns. Add 8 cups of water and one-half teaspoon salt, bring to a boil and reduce the heat. Simmer while you prepare the duck.
Step 2Use kitchen shears to cut the duck into 4 pieces (breasts and thighs). Reserve the liver. Cut the breasts in half crosswise if that will make them fit in the pan more easily or make it easier to have all the skin touching the pan.
Step 3Season the duck with salt and pepper and place the pieces skin side down in a large frying pan over high heat. You want all the skin touching the surface of the pan, so you may want to use two pans. Sear the duck skin for a few minutes until it begins to give off some fat. Add the garlic cloves and thyme sprigs to the pan and continue to cook the duck until the skin is crispy and brown, 10 to 12 minutes total. Drain off the excess fat, turn the duck pieces over and cook the flesh side just to sear, 3 to 4 minutes. Drain off all the fat and set aside.
Step 4Melt the butter with the olive oil in a large frying pan over medium-high heat. Add the diced carrots, celery and onion and sweat until the onion is tender and translucent, about 10 to 15 minutes. Season with salt and pepper.
Step 5Add the duck pieces and the Marsala and cook until most of the wine has evaporated, about 5 to 8 minutes. Add the tomatoes to the pan and break them up around the duck. (At this point, discard the garlic and thyme sprigs left in the bottom of the pan you cooked the duck in.) Add the juices from the tomatoes and simmer for about 5 minutes to cook off the rawness.
Step 6Add 4 to 6 cups of the simmering stock, pouring it through a strainer, until the duck is almost covered. Add the remaining bay leaf. Bring to a simmer over medium-high heat, reduce the heat, and cook the duck in the liquid at a low simmer for about 2 hours.
Step 7Remove the duck pieces from the sauce and remove and discard the bay leaf. Skim off the fat and set aside one-half cup of it. Any remaining fat can be reserved for another use.
Step 8Pass the sauce through a food mill or strainer, pressing the solids through, and return it to a clean pan. Pull the duck meat off the bones. You want to keep all the duck meat you can, as well as the nice clean slabs of duck skin from the breasts. Chop the duck meat and skin and add them to the pan with the sauce. Finely chop the liver and add it to the sauce as well.
Step 9Cook the meat in the sauce for about 20 minutes, until the sauce is thick and gravy-like. Stir in the reserved duck fat. Season with salt and pepper and toss with pappardelle. Top with grated cheese and parsley and serve while the pasta is still steaming.