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Place the chicken, onions, carrots, and celery in a large stock pot and cover with water. Bring to a boil, then reduce heat so water maintains a gentle simmer. Cook chicken for 1 hour or until cooked through (about 165 degrees).
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Once chicken is done, remove from the broth and let cool. Remove chicken from the bone and shred into medium-sized pieces, discarding bones and skin.
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Pour the chicken broth through a fine mesh sieve lined with cheese cloth, discarding vegetables. Reserve 6 cups of the broth for the dumplings. (You can refrigerate or freeze the rest to use in for another recipe.)
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Mix flour, baking soda, and salt together in bowl. Cut butter into flour mixture with your fingertips until it resembles small peas.
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Add milk — ¼ cup at a time, you may not need a full cup — and stir until a ball of dough just begins to form, being careful not to over-mix.
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Roll out the dough onto floured surface. Roll about ¼ inch thick. Using a pizza cutter or sharp knife, cut dough into rectangles about 1 inch wide by 3 inches long. Place strips on wax paper and allow to harden up a bit, about 30 minutes.
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In a large pot or dutch oven over medium-low heat, bring broth to a gentle simmer and drop in dumplings. Cover and allow to cook for 6-7 minutes.
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Reduce heat to low and add chicken. Allow to cook until thickened, approximately 15-20 minutes, or longer. Season generously with salt and pepper.