Combine chopped chicken, mayonnaise, egg, celery, salt, and pepper; mix well. Spread 1/3 cup chicken salad evenly on one side of 6 slices of bread. Place the remaining 6 bread slices over the chicken salad to make a sandwich.
Spread butter on outside of top slice of bread; invert sandwiches onto a hot skillet or griddle. Cook until bread is browned. Spread butter on ungrilled side of bread; carefully turn sandwiches, and cook until bread is browned.
Secure sandwiches with wooden picks; cut crosswise into 2 pieces. Serve hot. Garnish with whole olives, if desired.