Place the sliced onion and minced garlic in the bottom of the slow cooker.
Set the pork shoulder on top of the onion and garlic.
Sprinkle the smoked paprika, salt, and black pepper over the pork.
Pour the chicken broth around the sides of the meat.
Cover and cook on low for 8 hours until the pork shreds easily with two forks.
Remove the pork from the slow cooker and shred it on a cutting board.
Return the shredded pork to the slow cooker and stir in the barbecue sauce until evenly coated.
In a large bowl, combine the shredded cabbage, shredded carrot, mayonnaise, apple cider vinegar, sugar, and remaining salt and black pepper. Mix well.
Spoon the pulled pork onto the hamburger buns and top each with a generous scoop of coleslaw.