Prep and season: Preheat the oven to 325°F. Pat the pork dry with paper towels, then season liberally with salt and pepper on all sides.
Sear the roast: Heat the vegetable oil in a large Dutch oven (at least 7 quarts) over medium heat. Brown the pork on all sides, about 6-7 minutes per side, taking care not to let the oil get too hot. Remove the meat to a platter.
Cook the onions and garlic: Add them to the pot and cook in the drippings, stirring occasionally, until the onions are tender, about 10-15 minutes.
Add the liquid and herbs: Stir in 1½ teaspoons salt, the bay leaf, thyme, rosemary, apple cider, and chicken broth, scraping up any browned bits from the bottom of the pot.
Nestle the pork and cover: Settle the roast in fat-side up. The liquid should come about halfway up the sides. Put the lid on.
Braise until fork-tender: Roast in the 325°F oven for about 3½ to 4 hours, basting with the cooking liquid every hour or so if possible. The pork is done when it reaches an internal temperature of about 205°F and a fork twisted into the thickest part meets almost no resistance.
Shred the meat: Discard the bay leaf and herb stems. Transfer the pork to a large rimmed baking sheet or bowl. When it's cool enough to handle, shred with two forks, discarding any fatty pieces.
Serve: Spoon the cider cooking liquid and onions over the pork, or toss the shredded meat with warm barbecue sauce.