Place the pork shoulder in the slow cooker. Add the sliced onion, minced garlic, smoked paprika, and barbecue sauce on top.
Cover and cook on low for 8 hours until the pork is very tender and shreds easily with two forks.
Remove the pork from the slow cooker and shred it directly on a cutting board, then return it to the cooker and stir to coat with the juices.
In a separate bowl, combine the shredded cabbage, shredded carrot, mayonnaise, apple cider vinegar, sugar, salt, and pepper. Stir until the vegetables are evenly coated.
Split the sandwich buns and fill each with a portion of the pulled pork. Add a scoop of coleslaw on top or serve it on the side.