This recipe seems a little intimidating at first, mainly because of the process of making everything from scratch. But it's really important to do that! I've tried simpler ways, and the result is just not anywhere near as good. Trust me on this. Mostly, this is a method more than a recipe (you'll notice that's a theme with me) and it may take some trial and error. It's a very forgiving method, though!
1 broiler/fryer chicken, cut up
1 onion, peeled and quartered
3 cloves garlic, crushed and peeled
3 stalks celery, including leaves, cut into 3-inch pieces
2 carrots, peeled and cut into 3-inch pieces (or a handful of baby carrots)
salt and pepper, to taste (I think it's important to use a good amount of each!)
Place the chicken in a stockpot along with the vegetables and seasoning. Add water to just cover the chicken and veggies. Cover and bring to a boil. Remove lid and lower heat to med-low. Simmer, stirring occasionally, for 1 hour.
Remove chicken from the broth to a strainer over a large bowl. When cool enough to handle, de-bone chicken, removing skin as you go. While you are waiting for the chicken to cool, skim the vegetables out of your broth and discard them. Cut up or shred the chicken, placing it back into the broth, adding the broth that's collected in the bowl. Taste the broth and adjust seasoning. Bring the broth back up to a low boil.
Meanwhile, it's time to make "dumplin's", not to be confused with "dumplings" which might be made by a Yankee. (No offense, Sandy!)
I never measure my ingredients for dumplin's because Granny never does, either. Take it up with her! But here's what you need:
ice water, in an easy-to-pour container
Put the flour in a mixing bowl. With one hand, pour in the ice water a little at a time. With your other hand, (Yes: hand. Not spoon.) begin lightly mixing the flour and water. Keep going until the flour comes together with the water in a soft dough that is not sticky. Turn the dough onto a floured surface. Knead it 4-5 times, until it's a firmer, "rollable" texture. Divide the dough in half and roll one portion out into a vague circle. The dumplin's will puff up a good deal in the broth, so keep them thin. Using a sharp knife, slice the dough vertically into strips no wider than about 2 inches. Slice the dough once horizontally in the middle so they're not too long.
While stirring the broth with one hand, drop the dumplin's in one at a time with the other hand. When finished, immediately place the lid on the pot. Repeat the process with the other half of the dough.
If you have too little dumplin's when you're done, it's a snap to mix up more dough. If you have too many, well, there's no such thing as too many dumplin's!
Reduce the heat under the pot and cook the chicken and dumplin's, covered (very important!), for about 20 minutes. You may need to stir them once to prevent sticking. Taste and adjust the seasoning again if necessary.