Place the pork shoulder in the slow cooker. Add the chicken broth, 1/4 cup apple cider vinegar, brown sugar, paprika, garlic powder, onion powder, 1 teaspoon salt, and 1/2 teaspoon black pepper. Cover and cook on low for 8 hours until the meat shreds easily with a fork.

Remove the pork from the slow cooker and shred it with two forks. Return the shredded meat to the cooker and stir it into the juices. Let it rest on warm while you prepare the coleslaw.

In a large bowl, combine the shredded cabbage and carrot. In a small bowl, whisk together the mayonnaise, 2 tablespoons apple cider vinegar, Dijon mustard, sugar, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Pour the dressing over the cabbage mixture and toss until evenly coated.

Split the sandwich buns and toast them lightly if desired. Pile the pulled pork onto the bottom halves, then top each with a generous scoop of coleslaw. Close the sandwiches and serve right away.