Bring 6 quarts water to a boil and add 2 tablespoons salt.
Preheat oven to 400 degrees.
Cook pasta according to package instructions but remove 1 minute early. Drain and refresh with cold water and drain again and set aside. Place cool cooked sausage in food processor and pulse until ground but not much. Place in a large mixing bowl and add prosciutto cotto, eggs, 1/2 cup grated cheese and 1/2 cup chopped parsley and mix well. Season with salt and pepper and carefully stuff, using a piping bag or a small spoon, into pasta tubes. Layer pasta tubes in an oiled 8-inch by 10-inch baking dish, alternating with tomato sauce and grated cheese, until dish is full. Place in the oven and cook 15 to 20 minutes until golden brown and crispy on top and serve.
Saute the onion and garlic in the olive oil over medium heat until translucent, but not brown (about 10 minutes). Add the thyme and carrot and cook 5 minutes more. Add the tomatoes. Bring to a boil, lower the heat to just bubbling, stirring occasionally for 30 minutes. Season with salt to taste. Serve immediately, or set aside for further use. The sauce may be refrigerated for up to one week or frozen for up to 6 months.