Finger sandwiches are perfect for active parties, where guests may not sit down for a full meal.
Place chicken broth and the water in a medium saucepan. Add celery stalk pieces, carrot, onion, bay leaf, and peppercorns; bring mixture just to a boil. Reduce heat to a simmer, and add chicken breast. Poach until chicken is cooked through, 10 to 12 minutes. Transfer to a plate; let cool.
Transfer chicken to the bowl of a food processor fitted with the metal blade; process until it is finely ground. Transfer to a large mixing bowl. Add mustard, 1/2 cup mayonnaise, and remaining 1/2 cup diced celery. Season with salt and pepper, and stir well to combine. Cover with plastic wrap, and refrigerate until needed.
Make sandwiches: Spread a dab of remaining 1/2 cup mayonnaise on one side of each of the bread slices. Top half the slices with a thin layer of chicken salad and a few lettuce leaves; cover with remaining bread slices. Cut sandwiches in half. Stack several, and wrap well in plastic wrap. Refrigerate until ready to serve.