Heat the olive oil and prosciutto in a heavy 4-quart saucepan over medium-high heat. Add the onions. Saute the onions to a rich golden brown, about 10 minutes. Add the mushrooms and parsley, and saute until the mushrooms are a rich golden brown, about 10 minutes more. Add the basic tomato sauce and bring to a lively bubble. Cook, uncovered, for 5 minutes. Season, to taste, with salt and pepper. Add the cooked pasta to the pain and toss to coat well. Serve immediately, topped with grated cheese.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.