Cut chicken into quarters. Place into a large stock pot. Add water, celery, and onion. Cook 1 hour, or until chicken meat is falling off the bone. Remove the chicken from the pot, reserving the stock (broth) Strain reserved stock (broth) and return to the large stock pot. Cool chicken, then remove and discard bones, fat, and skin. Cut the chicken into bite sized pieces.
Dumplings: Combine flour, salt, and baking powder in a mixing bowl. Cut in shortening until it forms small crumbs. Add enough reserved stock (broth) to bring the dough to a soft ball. Roll out dough on a floured surface to approximately 1/8 inch or 1/4 inch thickness. Cut into 1/2 inch wide strips, 1 inch or 2 inches long
Over medium high heat return the reserved stock (broth), adding the carrots, back to a slow boil (do not boil rapidly). Gently stir the broth after adding several dumplings. Repeat until all the dumplings have been added, one at a time, to prevent sticking together. Continue stirring gently. Add the reserved chicken pieces, butter, and pepper to the dumplings. Turn heat to low, cover, and bring to a simmer. Simmer 10 minutes. Serve in a large soup bowl, or over mashed potatoes.