Combine garlic, oil, oyster sauce, vinegar, mustard, rosemary, thyme, pepper, salt and liquid smoke, if broiling chicken, in a small food processor or blender. Whirl until smooth puree. Transfer to saucepan; simmer for 4 minutes. Let cool.
Place vegetables in a plastic food-storage bag and the chicken in a second plastic bag. Divide marinade between bags; seal; refrigerate overnight, turning occasionally.
Prepare grill with medium-hot coals and hickory wood chips, or preheat broiler. Remove chicken and vegetables from marinade.
Grill vegetables and chicken, uncovered, 6 inches from heat, 16 to 18 minutes or until cooked through; turn chicken twice, vegetables three times. Or broil vegetables 4 inches from heat, 6 to 8 minutes per side. Place vegetables on platter; cover with foil. Broil chicken 4 inches from heat, 5 to 6 minutes per side or until cooked through.
Serve chicken with vegetables.
