01 - Place chicken breasts, chicken broth, barbecue sauce, apple cider vinegar, smoked paprika, garlic powder, onion powder, salt, and pepper in a medium saucepan. Bring to a simmer over medium heat, cover, and cook for 20–25 minutes until chicken is cooked through and tender.
02 - Remove the chicken from the pan and shred with two forks. Return shredded chicken to the pan, stir to coat with sauce, and simmer uncovered for 5 minutes until slightly thickened.
03 - In a large bowl, combine green cabbage, red cabbage, and grated carrot. In a small bowl, whisk together mayonnaise, apple cider vinegar, Dijon mustard, sugar, salt, and pepper. Pour dressing over the vegetables; toss until evenly coated. Chill until ready to use.
04 - Lightly toast the buns, if desired. Pile pulled chicken onto the bottom half of each bun. Top generously with slaw and pickle slices. Add extra barbecue sauce if desired, then cap with the top bun.