Adjust your oven racks so that your Dutch oven will fit inside. Preheat oven to 325°F.
Trim any large pieces of fat from roast and cut it into 4 evenly sized pieces. Generously salt and pepper all sides.
In a Dutch oven or large oven-safe pot with a lid, heat oil over medium-high heat. Brown pork on all sides and transfer to a plate.
If pork has rendered a lot of fat in the pot, spoon off all but 2 tablespoons and turn heat to medium-low. Add carrot, celery, and onion, and cook for 5 minutes.
Add garlic and cook until garlic begins to turn golden, 30 seconds to 1 minute.
Turn heat up to medium and add tomato paste. Cook until sugars in the paste begin to caramelize on the bottom of the pot, about 3 minutes.
Add red wine, and use a wooden spoon to scrape the bottom of the pot, removing anything stuck. Bring to a simmer, and cook until reduced by half, 3 to 5 minutes.
Stir in tomatoes, oregano, rosemary, thyme, and bay leaves. Return pork and any juices on the plate to the sauce and submerge. Bring to a simmer, cover and cook in the oven for 2 1/2 to 3 hours, until meat is pull-apart tender.
Put a pot of generously salted water on the stove to boil for pasta. Cook pasta according to package instructions.
While your pasta cooks, transfer meat to a plate or cutting board, and use two forks to shred, removing any fat or silver skin as you go.
Add half of the meat back to the sauce, and stir. Continue to add meat into the sauce until it reaches your desired sauce to meat ratio. If there is any shredded pork remaining, save it and use it for pulled pork sandwiches or topping pizza.
Taste sauce, and add salt if necessary.
Serve sauce over pasta, top with freshly grated parmesan, and enjoy!