While my Grandmother was visiting us from Missouri, I "picked her brain" as to how she makes her super-good chicken and dumplings. This is what we came up with.
Put one large, whole stewing chicken in your largest pot; cover with water. Add 1 whole onion for flavoring, peeled but not cut up. Add salt and pepper to taste, along with any other seasonings you like. Boil until the chicken is done, adding water as needed to keep the broth level high. Remove the chicken and onion from the pot. Discard the onion and cut up the chicken into an ovenware dish. Place the chicken in a warm oven until serving time.
Grandma said this is the tricky part. Dip out 2 cups of broth from the pot and put into a large bowl, then add 2 cups of water to the broth remaining in the pot. While the broth in the bowl is still hot, add enough flour a little at a time, stirring with a fork, to make a dough stiff enough to turn out on a floured surface. Knead the dough, adding flour as needed, until the mixture can be rolled.
Break off pieces of dough and roll until very thin. Cut these pieces into strips about an inch wide and 2 or 3 inches long. Bring the broth in the pot to a boil. As you cut the dumplings, drop a few into the boiling broth at a time, stirring often. Watch the dumplings carefully, because they burn easily. Dumplings should absorb most of the broth; they should be soupy not dry.
Avoid putting in more dumplings than you have broth for, or you may have better wallpaper paste than dumplings. Uncooked, leftover dough may be refrigerated for 4-5 days and used for dumplings with stews, ham and beans and other dishes.