Preheat oven to 325°F. Pat pork dry with paper towels; season liberally with salt and pepper on all sides.
Heat the vegetable oil in a large Dutch oven (at least 7 quarts) over medium heat. Brown the pork on all sides, taking care not to get the oil too hot, about 6-7 minutes per side. Remove the meat to a platter.
Add the onions and garlic to the pot. Cook, stirring occasionally, until the onions are tender, about 10-15 minutes.
Stir in 1 ½ teaspoons salt, bay leaf, thyme, rosemary, apple cider, and chicken broth. Nestle the pork shoulder in the pot (fat-side up). Cover the pot.
Roast the pork shoulder in the 325°F oven for about 3 ½ - 4 hours, basting with cooking liquid every hour or so, if possible. The pork is done when the meat reaches an internal temperature of 205°F. At this point, it should fall apart easily when pulled with two forks. Discard the bay leaf and herb stems. Transfer the pork to a large cutting board. When it's cool enough to handle, shred the meat with two forks (discarding fatty pieces). Spoon plenty of that delicious cider cooking liquid and onions over top, or toss the pulled pork with warm barbecue sauce.
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Notes
Alternate Slow Cooker Instructions: If you're going to be out of the house for most of the day, you can braise your pork shoulder in the slow cooker instead of in the oven. To do so, sear the pork according to the recipe instructions. Sauté the onions and garlic in the pot according to the recipe instructions, too. Deglaze the pot with the cider and broth. Transfer the onions and the liquid to the Crock Pot, stir in the remaining ingredients, and add the pork. Cover and cook on LOW for at least 8-10 hours or on HIGH for at least 4-5 hours.
If your pork shoulder is still tough at the end of the cooking time, it's probably because it needs to cook for even longer. The collagen in the meat is what keeps it tough, so you need to allow plenty of time for those fibers to break down. Just cover the pot and return the meat to the oven until it's fall-apart tender.
Add extra fresh herbs, spices, aromatics, and seasonings to the pot. For instance, try a couple sprigs of fresh oregano or fresh parsley, or season the meat with a bbq rub, our favorite house seasoning, coriander, cumin, or ginger. Cayenne pepper or crushed red pepper flakes will give the dish some heat, too!