Bring steaming water to a simmer under a steamer basket in a covered saucepan. Add chicken to the basket, cover and steam for 20 to 25 minutes until the thickest center reaches 165°F.
Meanwhile, place eggs in a second saucepan with egg-cooking water and bring to a boil. Cover, remove from the heat and let stand for 12 minutes, then transfer eggs to a bowl with the ice and cold egg-bath water for 5 minutes.
Shred cooked chicken into short pieces with clean forks and spread in a shallow container. Refrigerate promptly for about 60 minutes until 40°F or colder; the prep time includes this chilling period.
Peel cooled eggs and grate them on the coarse holes of a box grater. Mix mayonnaise, relish, mustard, celery, salt and pepper in a bowl, then fold in cold chicken and grated eggs.
Lay one dry lettuce leaf in each roll and divide chicken salad evenly among the 4 bottoms. Close gently and cut each sandwich across the middle.
Serve cold or refrigerate promptly at 40°F or below, packing with ice packs for transport.