Place chicken pieces in a very large stock pot, cover the chicken with water and place on the stove at high heat
Add carrot pieces, onion, and celery, 2 teaspoons of salt, and 1 teaspoon of pepper to the pot
While the chicken is cooking, add the flour, remaining 2 teaspoons of salt, and baking powder to a large mixing bowl
Cut in shortening with a fork or pastry cutter
Work the shortening in until there are no more clumps in the flour mixture
Add about ¼ cup of chicken stock to the beaten eggs (I typically use canned chicken broth for this step so I do not have to wait for the broth to cool down; it's an efficiency thing)
Add the eggs to the flour mixture and mix
Add the rest of the chicken broth to the flour mixture a little bit at a time until the dough becomes a little on the sticky side
On a large, clean surface, spread about 1-2 cups of flour in about a 2x4 foot area
Place ½ of the dough on the floured surface
Sprinkle a little flour on the top of the dough and press on it with your hands to flatten it a bit
Use a rolling pin (make sure it is well-floured), roll the dough out into a rectangle'ish
Fold half of the dough back on the other side of the dough and sprinkle the surface with more flour, repeat on the other side
Continue rolling the dough out until it is super thin (1/8 inch)
At this point, we need to check the chicken; when the chicken reaches 165℉, carefully remove it from the pot and place on a platter to cool
Remove the carrots, onions, and celery from the chicken stock
Keep the stock at a steady boil
Back to the dumplings; once the dumpling dough is as thin as you can get it, use a pizza wheel to cut 1 x 2-inch dumplings
Add the dumplings, one at a time, into the boiling chicken broth, stirring every once in a while, to keep the dumplings from sticking to the bottom and each other
Add more flour to the working surface and repeat the dumpling process with the remaining half of the dough
Once all of the dumplings are in the pot, reduce the heat so the dumplings are simmering, cover, and stir occasionally
While the dumplings are cooking, remove the chicken from the bone into bite-sized pieces, set the chicken meat aside and discard the bones
Once the dumplings are no longer doughy, taste to determine if any salt or pepper needs to be added
Add the chicken to the dumplings and stir
Serve over potatoes (boiled or mashed) and sweet peas