In a large pot, combine the chicken, chicken broth, onion, carrots, and celery. Bring to a boil, then reduce heat to a simmer. Cover and cook for 1-1.5 hours until the chicken is tender. Remove chicken and vegetables. Strain the broth through a fine-mesh sieve and return it to the pot.

Once cool enough to handle, shred the chicken meat, discarding bones, skin, and the cooked vegetables. Set the shredded chicken aside.

Bring the strained broth to a boil. Add the orzo and cook according to package directions until al dente (about 8-10 minutes). Reduce the heat to the lowest setting.

In a medium bowl, whisk the room temperature eggs for 2-3 minutes until pale and frothy. While still whisking, very slowly stream in the lemon juice.

Continuing to whisk constantly, slowly ladle about 2 cups of the hot broth from the pot into the egg-lemon mixture. This must be done gradually to avoid scrambling the eggs.

Slowly pour the tempered egg mixture back into the soup pot, stirring constantly. Add the shredded chicken. Let the soup gently heat for 5 minutes to thicken, but do not let it boil. Season with salt and pepper to taste and stir in the fresh dill before serving.