Whisk together bourbon, pineapple juice, hoisin sauce, ginger, 1 tablespoon pepper, Worcestershire sauce, garlic, and 1 teaspoon salt until blended. Place chicken in a shallow dish, add half the marinade and turn to coat. Cover and refrigerate overnight, turning occasionally. Cover and refrigerate remaining marinade.
Drain chicken, discarding the marinade. Grill chicken, covered, on a greased grill rack over medium-high direct heat until a thermometer reads 170°, about 5-6 minutes on each side.
Meanwhile, toss pepper and onion slices in olive oil, remaining 2 teaspoons salt, and 1 tablespoon pepper. Grill vegetables, turning frequently, until soft, about 5-7 minutes.
In a small saucepan on the grill, cook reserved marinade over medium heat, stirring occasionally, until slightly thickened, about 10 minutes. Chop grilled pepper and onion, sprinkle over chicken, and serve with sauce.