In a bowl, mix paprika, mustard, 2 Tbsp. brown sugar and 1 tsp. each salt and pepper. Add chicken and rub with spices to coat. Transfer to a slow cooker or cover and chill up to 8 hours to marinate.
In a medium bowl, whisk vinegar, ketchup, cayenne, remaining brown sugar and 1 tsp. each salt and pepper. Reserve 1/4 cup; cover and chill. Pour remaining sauce over chicken. Cover and cook on low until tender, 4 to 5 hours.
Toss reserved sauce with cabbage; cover and chill. Transfer chicken to a cutting board and let cool slightly. Discard bones and shred meat; transfer to a bowl. Carefully pour sauce from slow cooker into a fat separator. Pour skimmed sauce over meat; discard fat.
Divide meat among bun bottoms. Top with cabbage mixture and bun tops.