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Slowly stir in the milk, 1/4 cup at a time, just until a shaggy dough forms. You may not need the full cup of milk. Do not over mix the dough.
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Dump the dough out onto a floured surface and roll out to 1/4 inch thick. Use a pizza cutter to cut in long 1-inch wide strips. Cut the dough the other direction into 1-inch squares.
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Place dumplings on a parchment lined baking sheet to dry for 30 minutes.
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When the chicken is finished cooking (it should be falling off the bone) remove it to a large dish and, when cool enough to handle (or use forks), strip the meat from the bones and into small chunks.
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Pour the broth from the stock pot through a fine mesh sieve into a large bowl. Discard the vegetables. Return 6 cups of broth to the stock pot and refrigerate or freeze the remaining stock for another use.
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Bring the stock to a simmer and add the dumplings, one at a time so they do not stick together. Cook for 6-7 minutes.
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Add the chicken back to the stock pot and stir to combine. Continue cooking for 20 minutes. The broth should thicken as the dumplings cook.
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Add more salt and pepper to taste before serving.
