While the chicken is cooking and cooling, melt the butter in a large saucepan over medium heat. Add the onion and sauté until soft and translucent, about 5-7 minutes. Add the chili powder, garlic, and cayenne and cook until fragrant, about 30 seconds. Add the ketchup, apple cider, molasses, 3 tablespoons of the vinegar, Worcestershire sauce, mustard, and pepper; stirring until combined. Increase heat to medium-high and bring to a boil. Reduce heat to medium-low and simmer, uncovered, until flavors meld and the sauce is slightly thickened, about 15-20 minutes. Using a hand-held immersion blender, blend the sauce until desired consistency. You should have about 4 cups of sauce. Transfer 2 cups of sauce to a serving bowl; leave remaining sauce in the saucepan.