Place chicken in a shallow dish.
Combine soy sauce, sherry, honey, ginger, and garlic. Pour over chicken, turning to coat. Cover and marinate for at least 3 hours or up to 5 hours, turning occasionally.
Drain chicken, discarding marinade. Place the chicken on the unheated rack of a broiler pan. Place chicken on pan under broiler 4 to 5 inches from heat. Broil 10 to 12 minutes or until chicken is no longer pink, turning once.
Meanwhile, combine the mayonnaise and the Basil Vinegar.
To assemble the sandwiches, spread the mayonnaise mixture on warmed buns. Fill with lettuce, then chicken, onion, and tomato. Garnish with pickles. Makes 4 sandwiches.
In a non-metal container, place white vinegar or white wine vinegar and fresh basil. Cover and let stand in a cool, dark place for 24 hours.