For the chicken broth - Throw the leftover cooked chicken ( or raw chicken pieces ), bones and all, in a big pot with water and salt, enough to cover all the chicken. Let it cook at a slow boil until the chicken is falling off the bones. Then strain the broth off the chicken, return to heat and let it reduce until it has the "strength" of chicken flavor that you like. ( At this point, you can save the chicken broth to use at a later date. Freeze it if it will be more than a few days. )
For the dumplings:
1 cup unbleached flour
1.5 tsp baking powder
2 Tbsp oil
1/3 cup milk
1/2 tsp sea salt
Put flour, sea salt, and baking powder in a bowl, then add beaten egg, milk, and oil. Mix with a fork and let the dough sit for at least 15-20 minutes, covered.
Okay, once your dumpling dough is resting, start peeling and slicing carrots, slice celery, and dice onions; mince a little parsley, too, if you like, or maybe some peas. You can cook the veggies in the broth, or cook them separately in some salted water.
Bone some of the chicken, which should have cooled by now. If you prefer nice chunks of chicken instead of little pieces of leftovers, you can use a leftover cooked chicken breast or, maybe you can just use your roast chicken entirely for the soup, cutting off some of the best of the chicken before it goes to making broth. The boiled bits of chicken are good for chicken tacos.
Now, pat out your dumpling dough lightly on a lightly floured surface. Turn over once so they get flour on both sides. Cut with a biscuit cutter ( or inverted small glass ).
Make sure the veggies haven't overcooked. You can now steam your dumplings in a large pot of water that has been brought to a boil. You can steam them in the chicken broth, but I prefer to do it in a different pan of salted water that has been brought to a boil. ( This way your broth stays nice for the gravy. ) Reduce the heat to low (water should not be bubbling, but should still be very hot) and place dumplings in the hot water. Cover with a tight-fitting lid and let steam ever-so-slightly for about 7 minutes. You just want them to steam, not boil. After about 7 minutes, turn the dumplings over and cover again for about 5 minutes more.
Now turn you attention to the broth. Turn the heat up under the broth. In a cup, mix about 1/3 cup cornstarch in about a half to 3/4 cup of water. Stir until it has no lumps, then pour into hot broth while stirring all with a whisk. Add the cornstarch only about a third at a time, letting it get thick and adding more as you need it. When you have a nice gravy, add the chicken, veggies, and dumplings.