Simmer steaming water under a basket in a covered saucepan. Add chicken; cover and steam 20 to 25 minutes until thickest center reaches 165°F.
Meanwhile, boil eggs with egg-cooking water in a second saucepan. Cover, remove from heat, and rest 12 minutes. Chill in ice and cold bath water 5 minutes.
Shred chicken into short pieces with clean forks; spread in a shallow container. Promptly refrigerate about 60 minutes until 40°F or colder; chilling is included in prep time.
Peel cooled eggs; coarsely grate. Mix mayonnaise, relish, mustard, celery, salt, and pepper; fold in cold chicken and eggs.
Line each roll with one dry lettuce leaf; divide salad among 4 bottoms. Close gently; cut across middle.
Serve cold or promptly refrigerate at 40°F or below; transport with ice packs.