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Step 1
Remove the neck and innards (if included) from the chicken and discard. Place the chicken in the bottom of a large stock pot. Add the celery, onion, chicken base, salt, and pepper. Add enough water to just cover the chicken - about 12 cups. Place over medium heat and cook the chicken at a low boil for an hour or until the meat begins to fall off the bone.
Step 2
Remove the chicken from the stock and set aside. Remove the onion and celery from the pot with tongs, a strainer, or by pouring the stock through a fine mesh sieve.
Step 3
Make the dumplings by placing the flour in a medium bowl. Add the cold water and stir with a fork until combined. The dough will be shaggy. Use your hands to work the dough until it forms together. Turn the dough out onto a floured surface and knead a few times to ensure everything is combined. Dust flour under the dough and on top then use a rolling pin to roll the dough until it's about 1/8-inch thick. Turn the dough in a clockwise motion periodically to ensure it's not sticking to the surface. Add additional flour under or on top if necessary.
Step 4
Use a pizza cutter to cut the dough into 1-inch strips then cut each strip into about 2-inch lengths. Allow the dumplings to rest.
Step 5
Once cool enough to handle, remove the meat from the chicken and shred. Discard the bones and skin. Set the meat aside.
Step 6
Return the stock to medium-high heat. Add the cream of chicken soup and whisk to combine. Add additional salt and pepper gradually to taste. I've used as much as 1 additional tablespoon of salt in this step. It's important to get the stock seasoned well before adding the dumplings. If you prefer a thicker broth, whisk together 3 tablespoons of cornstarch with 1/4 cup of cool water to form a slurry. Whisk the slurry into the broth and bring to a boil to thicken.
Step 7
Bring the broth to a rolling boil. Add the dumplings to the stock one at a time - dropping them into the broth and not on top of one another. Stir gently. Once all of them are added, cook them for about 5 minutes or until they are tender and begin to float. Gently add the chicken to the pot and stir to combine. Cook long enough for the chicken to heat through.