Louisiana-style hot sauce gives the sandwiches a surprising kick, but feel free to substitute a sauce that is more fitting to your palate.
In a small saucepan over medium heat, bring the first five ingredients to a boil; boil, uncovered, for 1 minute. Cool for 10 minutes; set aside ¼ cup for basting. Flatten chicken to ½-inch thickness. Pour remaining marinade into a large resealable plastic bag; add chicken. Seal bag and turn to coat; refrigerate for at least 2 hours. Drain and discard marinade from chicken. Grill chicken, covered, over medium heat or broil 4 inches from heat for 5-6 minutes on each side or until no longer pink, basting occasionally with reserved marinade. In a small bowl, combine the blue cheese, buttermilk, mayonnaise, Parmesan cheese, chives, lemon juice, vinegar, garlic, and pepper. Spread over roll bottoms; top with the chicken, bacon, cheese, lettuce, tomato, and onion. Replace roll tops.