Melt butter in large saucepan over medium heat. Add onion; sauté until soft, about 6 minutes. Add next 5 ingredients, 1/4 teaspoon pepper, and chili powder. Reduce heat to medium-low; simmer until sauce thickens, about 15 minutes. Stir in bourbon; cook until heated through, about 3 minutes. Season with salt. (Can be made 1 day ahead. Cover and chill.)
Mix 1 1/2 teaspoons salt and 1/2 teaspoon pepper in bowl. Loosen skin on drumsticks. Rub salt and pepper mixture under skin without tearing skin. Cover; let stand at room temperature 30 minutes.
Prepare barbecue (medium heat). Grill drumsticks until skin is crisp and juices run clear, turning to cook all sides, about 25 minutes. Transfer 1/2 cup barbecue sauce to small bowl; reserve. Brush drumsticks with remaining sauce and cook until glaze forms, about 3 minutes longer. Transfer drumsticks to platter and serve with reserved sauce.
