Place the chicken, water, onion, carrots, celery, 1 1/2 teaspoons salt and 1/2 teaspoon pepper in a large pot and simmer gently for 70 to 80 minutes until the thigh reaches 165°F.
Lift out the chicken, strain the broth, discard the spent vegetables, and return 8 cups of broth to the pot.
Cool the chicken enough to handle, discard the skin and bones, and pull the meat into bite-size pieces.
Cut the shortening into the 2 1/2 cups flour, remaining salt and pepper until crumbly, then stir in the milk to make a firm dough.
Roll the dough 1/8 inch thick on a floured surface, cut it into 1-by-3-inch strips, and let the strips dry for 10 minutes.
Bring the broth to a gentle boil, drop in the dumplings one at a time, and simmer uncovered for 15 minutes until tender and no raw flour remains.
Fold in the chicken and heat for 5 minutes until it reaches 165°F and the broth lightly coats a spoon.
