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.)

In a large shallow bowl, cover the bread with the milk. Allow to sit for 10 minutes. Squeeze the milk from the bread by hand, cut the bread into a coarse dice and place in a medium bowl. Add the prosciutto and 1 cup Parmigiano-Reggiano, egg yolks, parsley, and salt and pepper and roll into balls the size of a quarter.

Roll each ball in the egg whites, then the breadcrumbs. In a heavy-bottomed skillet, heat the oil until almost smoking. Cook the balls until golden and remove to a plate lined with paper towels. When they?ve drained, divide them evenly among six warmed soup bowls, pour the hot broth over them and serve with the remaining Parmigiano-Reggiano sprinkled over the top.