Few recipes are more Southern or downright delicious than chicken and dumplings. Most folks won't get any more fancy with seasonings than parsley. The truth is that other fresh herbs, like chives, tarragon, or savory, are also good.
Place the chicken, onion, celery, and carrots in a 5-quart Dutch oven or covered casserole and add the water, ½ teaspoon salt, and ¼ teaspoon pepper. Bring to a boil over high heat, skimming off any foam that rises to the surface. Reduce the heat to low and cover tightly. Simmer, occasionally skimming the broth, until the chicken is tender, about 50 minutes.
Using tongs, transfer the chicken to a platter (keep the broth and vegetables simmering), and cool it until it's easy to handle. Discard the skin and bones and cut the meat into bite-size pieces. Meanwhile, increase the heat under the broth to high and cook until the liquid is reduced to 6 cups. (If you're in a hurry, strain the broth, reserving the vegetables, and measure 6 cups of broth, reserving the remaining broth for another use. Return the 6 cups of broth and vegetables to the pot). Skim off any fat from the surface of the broth. Stir the chicken back into the pot. Season to taste with salt and pepper.
To make the dumplings, place the flour, pinch of salt, and oil in a medium bowl and gradually stir in the water to make a stiff dough. Turn out onto a lightly floured surface and knead briefly. Roll out ¼ inch thick. Using a sharp knife, cut the dough into 1-inch wide strips. (If you wish, you can do what my mother does — make the dumpling strips while the chicken is simmering, and freeze the strips until ready to cook.)
Slide the strips into the simmering soup, placing them next to each other without stacking or crowding. Cover tightly and reduce the heat to low. Simmer until the dumplings are cooked through and tender, about 10 minutes. Sprinkle the dumplings with parsley. Serve from the pot, breaking up the dumplings as needed.