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PREHEAT greased grill to medium-high heat. Mix mayo and dressing. Spread 1/4 cup of the mayo mixture onto cut sides of bread. Brush half of the remaining mayo mixture onto the chicken and vegetable pieces.
PLACE chicken and vegetables, mayo-sides down, on grill grate. Grill chicken 16 min. or until cooked through (170°F) and vegetables 8 min. or until crisp-tender, turning and brushing chicken and vegetable pieces with the remaining mayo mixture halfway through the cooking times.
PLACE chicken on bottom half of bread; top evenly with vegetables. Cover with top of bread. Cut crosswise into eight pieces to serve.