Combine first 6 ingredients and mix until well combined. Reserve 1/4 cup and refrigerate. Pour the rest of the sauce in a ziplock bag and add the chicken breasts. Marinate overnight in the refrigerator.
Just before grilling (or broiling), drain and discard the marinade. Grill until juices run clear and chicken is thoroughly cooked.
On the bottom half of each roll, layer lettuce, tomato, chicken and green pepper. Drizzle with reserved sauce. Spread with mayonnaise if desired. Top with the rest of the roll.