Place the pork shoulder in the slow cooker. Add the sliced onion, minced garlic, paprika, salt, and black pepper.
Pour the barbecue sauce over the pork and onions. Cover and cook on low for 8 hours or until the pork is very tender.
While the pork cooks, make the coleslaw. In a mixing bowl, combine the shredded cabbage, shredded carrot, mayonnaise, apple cider vinegar, sugar, and salt and pepper to taste. Stir well and refrigerate until ready to serve.
Once the pork is done, remove it from the slow cooker and shred it with two forks. Return the shredded pork to the slow cooker and stir it into the remaining sauce.
Spoon the pulled pork onto the hamburger buns. Top each sandwich with a generous scoop of coleslaw.