Preheat the oven to 375 degrees . In a small bowl, mix 1/2 tbsp. of the sugar, 1 tsp. each salt and pepper, and the paprika.
In a large baking dish, place the chicken breasts, bone side down, and rub with the spice mixture; cover with foil. Roast until cooked through, 35 to 40 minutes. Transfer the chicken to a cutting board and let cool slightly (reserve the baking dish with the pan juices). Reduce the oven temperature to 275 degrees .
In a medium bowl, whisk the vinegar, ketchup, garlic, cayenne, if using, the remaining 2 tbsp. sugar and 1/4 tsp. each salt and pepper for the sauce.
Using 2 forks, shred the chicken; discard the bones. Transfer the chicken to the reserved baking dish. Add 1/2 cup of the sauce; toss to coat. Cover with foil and place in the oven until the chicken is heated through, about 20 minutes.
Toss the coleslaw mix with the remaining sauce. Pile the chicken on the buns. Top the chicken with the slaw or serve it on the side.