To a large pot, add the chicken, broth, water and vegetables. Bring to a boil over high heat, then turn down to medium. Partially cover the pot and cook until the chicken is falling off the bone and the vegetables are tender- about 30-45 minutes. Turn the heat to low.
Remove the chicken from the pot and set aside until it is cool enough to handle. Shred the chicken.
In a smaller pot, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes. Using a ladle, transfer approximately 2-3 cups of broth into the roux pot and whisk together. Bring to a simmer and cook until the mixture begins to thicken. Pour the thickened broth into the large pot, stir together, and allow to come back up to a simmer over medium heat. Whisk occasionally until the broth is smooth and thickened, 8-10 minutes. Stir in the reserved chicken, and turn the heat to low.
Meanwhile, add the flour, baking powder, parsley and salt to taste to the bowl of a food processor. Add the butter and pulse until the butter is evenly distributed. With the processor running, add the milk and egg, stopping as soon as the mixture comes together.
Using a spoon or a cookie scoop, drop heaping spoonfuls into the pot. Cover the pot and cook until the dumplings are fluffy, approximately 15 minutes. Serve at once.