1. In a large pot, combine the chicken, water, onion, carrots, and celery. Bring to a boil, then reduce heat and simmer for about 30 minutes, or until the chicken is cooked through.
2. Remove the chicken from the pot and set aside to cool. Strain the broth to remove the vegetables and return the broth to the pot.
3. Add the orzo to the broth and cook until tender, about 10 minutes.
4. While the orzo is cooking, shred the chicken meat, discarding the skin and bones.
5. In a medium bowl, whisk together the eggs and lemon juice until frothy.
6. Slowly add a ladle of hot broth to the egg mixture, whisking constantly to temper the eggs.
7. Gradually add the egg mixture back into the pot, stirring constantly. Do not let the soup boil.
8. Add the shredded chicken back to the pot and season with salt and pepper to taste. Serve hot.