Combine chicken, celery stalks, onion, carrot, peppercorns, allspice, bay leaf and salt in a large saucepan. Cover with water. Bring to a boil; reduce heat; simmer 2 to 3 hours or until chicken is tender.
Remove and discard vegetables. Allow chicken to cool in broth. Remove chicken. Skim off excess fat; strain broth. Remove skin and bones from meat. Cut meat into 1" pieces.
Melt butter or fat; add flour, stirring until smooth. Gradually stir ini the chicken broth and cook until it's thickened, stirring constantly. Add chicken and remaining ingredients. Simmer 5 minutes. Remove air bubbles. Adjust caps. Process pints 1 hour and 5 minutes, quarts 1 hour and 15 minutes, at 10 pounds pressure in a steam pressure canner.
Yield: about 4 pints or 2 quarts.
Brown 1-2 cups mushrooms in butter; add mushrooms to Chicken a la King, if desired. Heat until hot throughout.