In a large skillet over moderate heat, warm the olive oil. Add the garlic and rosemary and sauté until the rosemary needles sizzle and turn crisp and the garlic is golden, 2 to 3 minutes. Remove and reserve.
In the same pan, sear the pork in the hot flavored oil, turning it occasionally, until all sides are nicely browned, 5 to 7 minutes. Transfer to a large stockpot and set aside.
Add the onion to the flavored oil and cook, stirring occasionally, until browned (adding more oil sparingly if the pan is too dry), about 3 minutes. Crush the tomatoes with your hands and add, with their juice, stirring to scrape up any browned bits of pork stuck to the bottom of the pan. Transfer the tomato mixture to the stockpot and bring to a low boil over medium heat. Cover the pot, lower the heat, and simmer until the pork is tender enough to shred (when scraped with a fork), about 2 hours.
Cool the pork in the tomato sauce. (The cooled pork roast and sauce can be transferred to a storage container and refrigerated overnight. Wrap and store the garlic and rosemary separately.).
Remove the pork from the tomato sauce and reserve. In a food processor fitted with a metal blade, process half the tomato sauce with the fried garlic cloves and rosemary needles (discarding stalks) until somewhat smooth. Stir the puréed sauce back into the pot with the rest of the tomato sauce. The goal is a sauce with a rough, chunky character.
Shred the pork and strip the meat from the bones. Discard the bones. Chop the meat finely by hand, not in a food processor, and stir it into the tomato sauce. Heat and serve over pasta, passing the cheese on the side. Makes 8 cups, enough for 2 to 3 pounds pasta.