Combine brown sugar, smoked paprika, garlic powder, onion powder, cumin, salt, pepper, chili powder, and mustard powder in a small bowl.
Pat pork shoulder completely dry with paper towels. Trim fat cap to 1/4 inch if thicker. Coat all surfaces with dry rub, pressing firmly.
Preheat oven to 300°F (150°C) with rack in lower third.
Heat oil in Dutch oven over medium-high heat. Sear pork on all sides until deep golden-brown, 3-4 minutes per side. Transfer to a plate.
Add chicken broth to the pot and scrape up browned bits. Stir in BBQ sauce and apple cider vinegar, bringing to a gentle simmer.
Return pork to the pot, nestling into liquid (should come 1/4 to 1/3 up the sides). Cover tightly and braise in oven for 3.5-4.5 hours.
At the halfway point, flip the pork in the braising liquid, re-cover, and continue cooking until internal temperature reaches 195-205°F (90-96°C).
Transfer pork to a cutting board and rest for 15-20 minutes. Shred with two forks, discarding bone and large fat chunks.
Strain braising liquid and skim off fat. Drizzle over shredded pork, tossing to moisten. Adjust seasoning with BBQ sauce, salt, or vinegar to taste.