Place the pork shoulder in the slow cooker. Add the sliced onion, minced garlic, chicken broth, smoked paprika, salt, and black pepper. Cover and cook on low for 8 hours until the meat is very tender.
Remove the pork from the slow cooker and shred it with two forks. Return the shredded meat to the slow cooker and stir in the barbecue sauce. Keep the cooker on warm while you prepare the coleslaw.
In a large bowl, combine the shredded green cabbage and grated carrot. In a small bowl, whisk together the mayonnaise, apple cider vinegar, sugar, and celery seeds until smooth.
Pour the dressing over the cabbage mixture and toss until everything is evenly coated. Season with a pinch of salt and pepper if needed.
Serve the pulled pork topped with a generous scoop of the tangy coleslaw.