01 - Preheat the oven to 325°F.
02 - Pat pork shoulder pieces dry and season generously with salt and black pepper.
03 - Heat olive oil in a large Dutch oven over medium-high heat. Sear pork pieces until browned on all sides, approximately 6-8 minutes. Transfer seared pork to a separate plate.
04 - In the same Dutch oven, add diced onion, carrot, and celery. Sauté for 4-5 minutes until softened. Stir in minced garlic and cook for 1 minute until fragrant.
05 - Return the seared pork to the Dutch oven. Add apple cider, chicken broth, cinnamon stick, fresh thyme sprigs, and bay leaf. Bring the liquid to a simmer, ensuring to scrape any browned bits from the bottom of the pot.
06 - Cover the Dutch oven and transfer it to the preheated oven. Braise for 2 hours, or until the pork is exceptionally tender and easily shreddable.
07 - Remove and discard the cinnamon stick, thyme sprigs, and bay leaf from the braised pork. Using two forks, shred the pork into bite-sized pieces. While the pork finishes braising, cook the pasta in a large pot of generously salted boiling water until al dente. Reserve 1/2 cup of the pasta cooking water before draining the pasta.
08 - Place the Dutch oven containing the braised pork over medium heat. Stir in the unsalted butter and, if using, the heavy cream until melted and incorporated. Add the drained, cooked pasta to the Dutch oven and toss to combine with the pork and sauce. Add reserved pasta water as needed to achieve a desired sauce consistency.
09 - Stir in the grated Parmesan cheese until it is fully incorporated into the sauce. Taste and adjust seasoning with additional salt and pepper as necessary.
10 - Serve the pork pasta hot, garnished with extra grated Parmesan cheese and chopped fresh parsley.