In a large pot or Dutch oven, combine the chicken, chicken broth, onion, carrots, and celery. Bring to a boil, then reduce heat to a simmer. Skim any foam from the surface. Cook for 45-60 minutes, or until the chicken is cooked through and tender.

Carefully remove the chicken from the pot and set it aside to cool. Strain the broth through a fine-mesh sieve into a clean pot; discard the solids. You should have about 8 cups of broth.

While the chicken cools, bring the strained broth to a simmer over medium heat. Add the rice or orzo and cook according to package directions, until tender (about 15-20 minutes for rice, 8-10 for orzo). Reduce heat to low.

Once cool enough to handle, shred the chicken, discarding the bones and skin.

In a medium bowl, whisk the eggs until pale and very frothy, about 2-3 minutes. While whisking continuously, slowly stream in the fresh lemon juice.

To temper the eggs, slowly ladle about 2 cups of the hot broth from the pot into the egg-lemon mixture, one ladleful at a time, while whisking vigorously. This is crucial to prevent the eggs from scrambling. The egg mixture should now feel warm.

Pour the warm egg-lemon mixture back into the soup pot in a slow, steady stream, stirring the soup constantly. Add the shredded chicken to the pot.

Continue to cook over low heat, stirring gently, for 3-5 minutes until the soup has thickened slightly. Do not let it boil. Remove from heat and stir in the fresh dill and parsley. Season generously with salt and pepper to taste.