Heat oil in a large, heavy pot over medium heat. Saute onions, celery, carrots and garlic until tender, about 10 minutes. Increase heat to high and add veal, pork, sausage and pancetta; cook until meats are evenly brown.
Stir in tomatoes, chicken broth, milk, parsley, basil and thyme. Reduce heat to medium-low and simmer, uncovered, for 2 1/2 hours; stir occasionally, breaking up tomatoes with spoon as you stir. Season with salt and pepper to taste.
Bring a large pot of lightly salted water to a boil. Add fettuccini and cook for 8 to 10 minutes or until al dente; drain.
Serve fettuccini topped with the sauce and Parmesan cheese.