Poach the chicken in the broth with the rib of celery, the carrot, onion, black pepper, and one-eighth teaspoon cayenne pepper until tender, 30 to 40 minutes. Salt to taste; cool in the broth; debone and chop fine.
Mix the chicken with 1 cup mayonnaise. Add the lemon juice, one-fourth teaspoon of the cayenne, the chopped celery and salt to taste. Fold in the eggs and add additional mayonnaise if you wish. Cover and refrigerate several hours or overnight.
Yield: enough spread for approximately 15 to 20 sandwiches, quartered