In a large Dutch oven or stockpot, add the chicken, the carrots, the celery, and the onion.
Add 4 quarts of the water, the vinegar, and 2 teaspoons of the salt to the pot, adding the extra water as needed to ensure the chicken is fully submerged.
Bring the chicken mixture to a boil.
Reduce the heat to low.
Cover the pot and let the mixture simmer until the meat nearly falls off the bones and reaches an internal temperature of 165 degrees F, about 1 hour 30 minutes.
Transfer the chicken from the broth and allow it to cool.
Strain the broth into a bowl and discard the collected vegetables.
Return the strained broth to the pot.
Set aside 1 cup of the broth and allow it to cool slightly.
Shred the meat from the bones and discard the skin and bones.
Cut the meat into bite-size pieces, set it aside, and keep it warm.
In a bowl, add 2 cups of the flour and the remaining salt and whisk to combine.
Make a well in the flour mixture and add the egg to the well.
Gradually stir 1/4 cup of the reserved broth into the egg, picking up and incorporating the flour mixture while stirring. Continue until the flour is all incorporated, adding more of the reserved broth as needed until the dough resembles the consistency of pie dough. Return any of the remaining broth back into the stockpot.
Continue to add the reserved chicken broth to the well and incorporate the flour mixture gradually as you stir until all of the flour is incorporated and the dough resembles the consistency of a pie dough. You may not need all of the reserved broth; return any remaining broth to the pot.
Using the extra flour, dust a work surface.
Turn the dough out onto the prepared work surface and knead the dough until stiff.
Let the dough rest for 15 minutes.
Return the dough back to the floured work surface, adding more of the extra flour as needed, and roll out the dough into a 17-inch square.
Cut the dough square into 1-inch pieces and dust the dough pieces with the extra flour.
Let the dough pieces dry, about 30-60 minutes.
Bring the remaining broth to a boil on the stovetop.
Carefully drop the dumplings into the boiling broth.
Reduce the heat to low and cover the pot.
Simmer the dumplings until a toothpick inserted into the center of a dumpling comes out clean, about 10 minutes.
Add the reserved chicken and the pepper to the pot and stir to combine.
Serve garnished with the fresh herbs.