Place the chicken breasts in the slow cooker. Sprinkle them with smoked paprika, garlic powder, salt, and pepper.
Pour the chicken broth and 1 cup of BBQ sauce over the chicken. Stir gently to coat.
Cover and cook on low for 6 hours until the chicken reaches an internal temperature of 165°F and shreds easily.
Remove the chicken, shred it with two forks, and return it to the slow cooker. Stir in the remaining 1/2 cup of BBQ sauce and let it warm for 10 minutes.
While the chicken finishes, combine the shredded cabbage, shredded carrot, mayonnaise, apple cider vinegar, and sugar in a bowl. Mix until the vegetables are evenly coated.
Spoon the chicken onto the hamburger buns and top each sandwich with a generous portion of slaw.