Preheat the oven to 425 degrees . Place the chicken on a rimmed baking sheet and season with salt and pepper. Roast until the juices run clear, about 45 minutes. Let cool and then shred; discard the skin and bones.
Meanwhile, place the potatoes in a medium saucepan with enough cold water to cover by 1 inch. Bring to a boil, then simmer until, about 45 minutes. Drain.
In a large pot, melt the butter over medium heat. Add the onion and cook, stirring often, until softened, about 8 minutes. Pour in the broth and bring to a gentle simmer.
Meanwhile, mash the potatoes with 1 teaspoon salt until almost smooth. Mix in the egg, then the flour, until combined.
On a lightly floured surface, roll one-third of the dough into a 3/4-inch thick log. Pinch off inch-long pieces and roll to form small disks. Repeat with the remaining dough.
Add the dumplings to the broth. Once they float, cook for 3 to 4 minutes more. Stir in the reserved chicken and season with salt and pepper. Top with the parsley.