Place the pork shoulder in the slow cooker and scatter the sliced onion and minced garlic around it.
In a small bowl stir together the barbecue sauce, 2 tablespoons of the apple cider vinegar, brown sugar, smoked paprika, salt, and pepper. Pour this mixture over the pork.
Cover and cook on low for 8 hours until the pork shreds easily with two forks.
While the pork cooks, make the slaw. In a medium bowl combine the shredded cabbage, shredded carrot, mayonnaise, Dijon mustard, and the remaining 2 tablespoons apple cider vinegar. Toss until evenly coated and set aside.
When the pork is done, shred it directly in the slow cooker and stir it into the sauce.
Split the buns and spoon the pulled pork onto each bottom half. Top with a generous scoop of the tangy slaw and close the sandwiches.