boneless pork shoulder or butt, trimmed and cut in half
grated Parmesan, for serving
In a 4- to 6-quart slow cooker, combine the carrot, onion, garlic, tomato paste, thyme, oregano, ¾ teaspoon salt, and ¼ teaspoon pepper. Add the tomatoes (and their juices); add the pork and turn to coat.
Cover and cook until the pork is very tender, on low for 7 to 8 hours or on high for 5 to 6 hours (this will shorten total recipe time).
Twenty minutes before serving, cook the fettuccine according to the package directions; drain and return it to the pot. Meanwhile, using 2 forks, shred the pork and mix it into the cooking liquid. Toss the pasta with the ragù and sprinkle with the Parmesan.