For slaw, in a medium bowl whisk together the mayonnaise, sour cream, vinegar, ½ tsp. of the salt, the black pepper, and celery seeds.Add cabbage, carrot, and the 1/3 cup sliced red onion; toss to coat . Cover and chill at least 1 hour before serving.
Meanwhile, pull apart mushrooms using a fork. Heat a large skillet heat oil over medium-high heat. Add chopped onion. Cook, stirring occasionally, until onion is just seared at the edges. Add mushrooms. Sprinkle with the remaining ½ tsp. salt. Cook, stirring occasionally, until excess liquid evaporates and mushrooms are tender. Add bell pepper and paprika. Cook, stirring occasionally, about 5 minutes or until peppers are tender. Stir in BBQ sauce. Bring to a simmer. Cook and stir about 5 minutes more or liquid has evaporated and mixture is until very thick. Remove from heat.
Preheat broiler. Divide mixture among bun bottoms. Top with cheese. Broil about 8 inches from the heat about 5 minutes or until cheese is melted. Top with slaw and bun tops.