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.