Mix liquid from pickled vegetables, oil and mustard in small bowl. Reserve 1/4 cup of the marinade for basting. Place chicken in large resealable plastic bag or glass dish. Add remaining marinade; turn to coat well. Refrigerate 2 hours or longer for extra flavor.
Meanwhile, mix mayonnaise and roasted ginger in small bowl until well blended. Cover. Refrigerate until ready to serve. Remove chicken from marinade. Discard any remaining marinade.
Grill chicken over medium-high heat 4 minutes per side or until cooked through, turning occasionally and brushing with reserved marinade. Cut rolls in half lengthwise. Spread with mayonnaise mixture. Serve chicken in rolls topped with Asian-Style Pickled Vegetables.