Place the chicken breasts in the slow cooker and add the sliced onion on top.
Stir together the barbecue sauce, chicken broth, smoked paprika, and garlic powder in a small bowl, then pour the mixture over the chicken.
Cover and cook on low for 6 hours until the chicken is tender enough to shred easily.
While the chicken cooks, combine the shredded cabbage and carrot in a medium bowl.
Whisk the mayonnaise, apple cider vinegar, salt, and black pepper in a separate small bowl, then pour it over the cabbage mixture and toss until evenly coated. Cover and refrigerate the coleslaw until serving.
Once the chicken is done, remove it from the slow cooker and shred it with two forks. Return the shredded chicken to the slow cooker and stir it into the remaining sauce.
Slice the sandwich rolls and fill each one with a portion of the pulled chicken, then top with a scoop of coleslaw.