Remove chicken pieces, if any, from inside the chicken cavity. Discard and place chicken in a large Dutch oven. Add water to the Dutch oven to cover or almost cover the chicken. Cook until the chicken is completely done.
Remove chicken from the pot and set aside to cool.
Strain broth in the Dutch oven into a large measuring cup. Add 8 cups of chicken broth back to the dutch oven, set aside. (If you don't have 8 cups of broth add enough homemade broth or canned broth to equal 8 cups.
Mix 2 1/2 c. all-purpose flour and 1 tsp. salt in a food processor. Pulse for a few seconds to combine. With the motor running, pour in ice water a little at a time,just until dough starts to come together. Dump onto a floured board. Roll the dough to 1/8" thickness or less. Cut into 1-inch strips (about 3 to 4 inches long). Let dry while you prepare broth.
Combine chicken broth, 4 c. water, 1 tsp. salt, 1/2 tsp. black pepper, and 4 tbsp. butter in the Dutch oven.
Pick cooked and cooled chicken off the bones and add to the pot. Bring to a boil.
Drop dumplings into boiling broth. After you have added the dumplings, place the lid on the pot and reduce heat to medium. Cook for about 10 minutes, stirring occasionally. Reduce heat to a simmer, leaving the lid on, and continue to cook, stirring occasionally, for about 20 minutes.
