Preheat the oven to 350°F and grease a 9×13-inch baking dish.
In a large bowl, mix shredded chicken, BBQ sauce, ½ teaspoon salt, and ½ teaspoon pepper.
In another bowl, combine green cabbage, purple cabbage, carrots, paprika, sour cream, lemon juice, mayonnaise, and the remaining salt and pepper. Mix well.
Slice the rolls horizontally and place the bottom half in the prepared baking dish.
Spread the BBQ chicken mixture evenly over the rolls, followed by the slaw. Place the top half of the rolls over the filling.
Melt the butter and brush it over the top of the rolls.
Bake uncovered for 13 to 15 minutes, until the buns are golden brown and heated through.
Remove from the oven and serve warm.