Instead of chicken quarters, you can use meaty chicken pieces. Whichever you choose, the honey-mustard-bourbon marinade will give a sweet mellow flavor to grilled chicken.
Prepare outdoor grill with medium coals, or heat gas grill to medium (to broil, see Note below).
Whisk mustard, bourbon, oil, garlic, 2 tablespoons honey, salt, pepper and paprika in cup. Remove skin from chicken. Combine chicken with marinade in plastic food-storage bag or glass dish; seal or cover with plastic wrap. Refrigerate 30 minutes.
Remove chicken from marinade; transfer marinade to small bowl.
Grill chicken, turning frequently, 35 to 40 minutes (internal breast temperature registers 170°F); for first 20 minutes, baste chicken occasionally with marinade. Transfer remaining marinade to saucepan.
Mix broth and cornstarch in bowl.
Whisk remaining 1 tablespoon honey and brown sugar into marinade in saucepan. Boil 3 minutes. Stir in cornstarch mixture; boil 1 minute.
Transfer chicken to platter. Drizzle with sauce. Serve with remaining sauce.
Broil 6 inches from heat for 20 minutes, turning frequently.