Heat the oil in a large pot (preferably a Dutch oven) over medium-high heat. Sauté pancetta for 5 minutes.
Add the onions, celery, and carrots. Cook until soft, about another 10 minutes.
Add beef, pork, and veal and sauté, breaking it up, until browned, about 15 minutes.
At this point, if you feel you have too much fat that has rendered, tilt the pot and use a large spoon to collect the fat that has pooled at the end of the pot. Remove all grease except for about 1 tbsp.
Add wine; boil for 1 minute, deglazing the pan (scraping the brown bits from the bottom of the pan).
Add 2½ cups of the broth and the tomato paste; stir. Use your hands to squeeze and gently break the whole tomatoes into the sauce. Reduce heat and gently simmer, stirring occasionally, 1½ hours. Season with salt (2 tsp) and black pepper (1 tsp).
Warm the milk in a small saucepan and then gradually add to the sauce.
Simmer the sauce for another 30 minutes, stirring occasionally, until thickened and milk has completely been absorbed. Add more broth if the sauce is too thick. Taste and add more salt, if necessary.
Meanwhile, cook pasta in liberally salted, boiling water until al denté. Follow packaging instructions. Reserve ½ cup of the pasta liquid. Drain pasta.
Transfer 2 large spoonsful of the sauce to a large skillet over medium-high heat. Add pasta and toss until coated. Add in remaining sauce and a few tablespoons of the reserved pasta water until pasta is fully coated and the consistency is right.
Serve at once with Parmesan cheese on the side.