In a small bowl, combine the brown sugar, smoked paprika, garlic powder, onion powder, chili powder, cumin, salt, black pepper, and ground mustard. Mix until fully combined.
Pat the pork shoulder dry with paper towels. Cut into 3 to 4 equal-sized chunks and sprinkle the dry rub generously over all sides, pressing it into the meat with your hands to adhere.
In the inner pot of the Instant Pot, pour in the broth, apple cider vinegar, and liquid smoke. Stir briefly to combine.
Place the seasoned pork chunks into the braising liquid. Close the lid, set the valve to "Sealing," and pressure cook on HIGH for 60 minutes.
When the timer is done, let the pressure release naturally for 15 minutes, then carefully turn the valve to "Venting" to release any remaining pressure.
Transfer the cooked pork to a large bowl and shred it with two forks or meat claws. Discard any large pieces of fat or connective tissue that did not render down.
Drizzle 2 to 3 tablespoons of the reserved cooking liquid over the shredded pork and toss to coat. Add the BBQ sauce and toss well to coat every strand.
Optionally, spread the sauced pork on a foil-lined baking sheet and broil at 425°F (220°C) for 5 to 7 minutes to caramelize the sauce and add crispy edges.