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To make the gravy:
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In a large (8- to 10-quart), heavy-bottomed soup pot or Dutch oven, melt the butter over medium heat.
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Add the flour and mix well to make a roux.
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Cook, stirring frequently, until the mixture is pale yellow and resembles fine, wet sand, about 3 minutes.
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Whisk the broth into the roux a little at a time, allowing the roux to absorb the liquid before adding more (this will help prevent lumps).
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Add the salt, pepper, carrots, and celery.
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Bring the mixture to a boil over high heat, stirring occasionally, and then lower the heat and gently simmer for 15 to 20 minutes.
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Keep stirring occasionally and skim off any scum (Love Note 2) that rises to the surface.
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To make the dumplings:
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Whisk the flour, baking powder, salt, and parsley together in a large bowl, and then cut in the butter using a pastry blender, two knives, or a whisk until its in small pieces.
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(Alternatively, you can use a food processor: Place the flour, baking powder, and salt in the bowl of a food processor.
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Pulse several times to combine.
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Add the parsley and pulse once or twice to incorporate.
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Add the butter and pulse until it is in small pieces.)
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Add the milk and stir or pulse once or twice to moisten the flour mixture.
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(Do not overmix or you will develop the gluten in the flour and the dumplings will be chewy.)
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Gather the dough into a ball.
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Using a large spoon or your hands, scoop out 1/4-cup chunks of dough, lightly roll them between your palms to round them out, and then drop into the simmering gravy (its ok if they sink), spacing them apart.
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Cover the pot and simmer until the dumplings are done (a knife inserted in the center should come out clean), about 20 minutes.
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(Avoid lifting the lid while the dumplings are cooking because it slows down the cooking process, and if youre lookin youre not cookin!)
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To serve:
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Gently stir the cooked chicken into the pot with the dumplings, return the liquid to a simmer, cover, and cook for 5 more minutes to heat the chicken through.
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Using a serving spoon or tongs, divide the chicken and dumplings among soup bowls.
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Ladle the gravy over the dumplings and chicken, sprinkle with the parsley, and serve.
