Set a large pot of water on the stove to boil, and cook the fettuccine according to the directions on the box.
In the meantime, heat the oil in a large skillet over a medium-high heat. Add the onion, carrots, and celery and cook for 5 minutes, stirring occasionally. Add the mushrooms and cook for 5 minutes more, stirring occasionally. Add the garlic, ground beef, and thyme and cook until the meat is browned, breaking it up into small pieces as it cooks, about 5 minutes.
Stir in the tomatoes and broth and cook for 5 minutes, or until sauce is thickened. Stir in the milk and cook for 1 minute more. Season with salt and pepper.
When the pasta is done, drain it, put it back in the pasta pot, add the sauce and stir the sauce and pasta together. Put into serving bowls and top with Parmesan cheese.