Season the pork: Stir the brown sugar, smoked paprika, salt, chili powder, garlic powder, onion powder, ground mustard, and black pepper together in a small bowl. Pat the pork completely dry with paper towels and rub the mixture over every surface, pressing it in. Let it sit at room temperature for 30 minutes, or wrap and refrigerate overnight for deeper seasoning.

Sear it hard: Heat the oven to 300°F. Heat the vegetable oil in a large Dutch oven over medium-high heat until it shimmers. Sear the pork for 3-4 minutes per side, turning until all sides have a deep brown crust. Move it to a plate.

Start the braise: Add the sliced onion to the pot and cook for 3 minutes, stirring, until it picks up some color. Pour in the chicken broth and apple cider vinegar and scrape the browned bits off the bottom of the pot with a wooden spoon.

Roast low and slow: Return the pork to the pot, cover tightly, and roast for 5 to 5 1/2 hours. It's done when a thermometer in the thickest part reads 200-205°F and a fork twisted into the meat turns with almost no resistance. If it's at 190°F and still feels firm, put it back for another 30 minutes.

Rest before shredding: Move the pork to a cutting board or large bowl and let it rest for 25 minutes. Meanwhile, let the cooking liquid settle and spoon the fat off the top, saving the seasoned juices underneath.

Shred it: Pull the pork apart with two forks, discarding any large pockets of fat as you go. It should come apart in long strands with very little effort.

Sauce and serve: Toss the shredded pork with about 1 cup of the skimmed cooking juices, then stir in the barbecue sauce until it's coated the way you like. Pile it onto hamburger buns and pass extra sauce at the table.