Place the chicken into a large pot. Cover with water and bring to a boil. Add the onion, carrot and celery. Simmer slowly until tender, about 2 ½ to 3 hours. Remove the chicken and debone, then chop into small strands. Set aside. Save ½ cup of chicken broth and let cool. With an immersion blender, puree all the vegetables until they are liquid form. (If you don’t have an immersion blender, you can strain and save the broth by separating the vegetables into a sieve or colander, then put the vegetables into a regular blender and puree. Put them and the broth back into the pot you cooked the chicken in.) Season the remaining broth with butter and salt and pepper to taste. Bring to a boil and cook the dumplings.
In a mixing bowl, cut the shortening into the flour and stir in the cooled broth, cover and let stand about 30 minutes. Roll out as thin as possible on a floured pastry cloth. Cut into strips or small squares. Drop layers of dumplings into the boiling broth and let the broth boil over each layer before adding another layer. Do not stir. Cover and cook for 12 minutes. Carefully add chicken which has been deboned and chopped.