Heat oil in a large Dutch oven over medium heat. Add onion, celery, and carrot; cover and cook 8 minutes, stirring occasionally. Remove onion mixture from pan.
Add veal, pork, and beef to pan; cook over medium heat until browned, stirring to crumble. Add wine, salt, pepper, nutmeg, and bay leaf; bring to a boil. Cook 5 minutes. Add the onion mixture, broth, and tomato puree; bring to a simmer. Cook 1 hour, stirring occasionally.
Stir in milk and minced parsley; bring to a boil. Reduce heat, and simmer 40 minutes. Discard bay leaf. Add pasta, and toss to coat. Sprinkle evenly with cheese. Garnish with parsley sprigs, if desired.