Pat the pork dry with paper towels and season with salt, pepper, and smoked paprika.
In a bowl, whisk together apple cider vinegar, garlic powder, and onion powder.
Submerge the pork in the cure, ensuring it’s fully covered. Refrigerate uncovered for 6–8 hours or overnight.
Bring the pork to room temperature for 1 hour before cooking.
Heat olive oil in a Dutch oven over medium-high until shimmering.
Sear the pork on all sides, 2 minutes per side, until golden and caramelized.
Add enough chicken or vegetable broth to the Dutch oven to cover about one-third of the pork.
Reduce the heat to low, cover, and simmer for 2.5 hours, or until the pork is tender and easily shreddable with a fork.
Shred the pork into small pieces using two forks.
Add barbecue sauce and stir until evenly coated.
Toast the buns lightly if desired.
Assemble each sandwich with a portion of pork, coleslaw, pickles, and red onion slices.
Optionally drizzle buns with balsamic glaze (1 part honey to 2 parts balsamic vinegar). Serve immediately.