Phase 1: Creating the Flavorful Chicken and Broth

1. Prepare the Chicken: Rinse the whole chicken or chicken pieces under cold water. If using a whole chicken, remove any giblets from the cavity (you can add these to the pot for extra flavor if desired, then remove later).
2. Simmer the Broth: Place the chicken, quartered onion, chopped carrots, celery stalks, bay leaves, and peppercorns into a large stockpot or Dutch oven. Pour in the cold water, ensuring the chicken is mostly submerged. Add an initial teaspoon of salt.
3. Bring to a Boil, then Simmer: Bring the water to a boil over medium-high heat. Once boiling, you’ll notice foam and impurities rising to the surface. Skim this off with a large spoon or skimmer for a clearer broth. Reduce the heat to low, cover the pot loosely, and let it simmer gently for 1 to 1.5 hours, or until the chicken is cooked through and tender. For bone-in pieces, this might take closer to 45-60 minutes. The chicken should be easily falling off the bone.
4. Remove Chicken and Strain Broth: Carefully remove the cooked chicken from the pot and set it aside on a cutting board to cool slightly. Strain the broth through a fine-mesh sieve into a clean pot or large bowl. Discard the solids (onion, carrots, celery, bay leaves, peppercorns). You should have about 8-10 cups of rich chicken broth. If you have significantly more, you can simmer it down a bit to concentrate the flavor. If less, you can top it up with a little water or store-bought chicken broth.
5. Shred the Chicken: Once the chicken is cool enough to handle, remove the skin and bones. Shred or dice the meat into bite-sized pieces. Set aside.

Phase 2: Cooking the Soup Base

6. Cook the Rice/Orzo: Return about 8 cups of the strained chicken broth to the stockpot (reserve any extra broth for later if needed). Bring the broth to a gentle simmer over medium heat. Add the rice or orzo and cook according to package directions, or until tender. For rice, this is typically 15-20 minutes. For orzo, it’s usually 8-10 minutes. Stir occasionally to prevent sticking.
7. Add Chicken Back: Once the rice or orzo is cooked, stir the shredded chicken back into the soup. Reduce the heat to very low to keep the soup warm while you prepare the avgolemono sauce. It’s crucial the soup is not boiling when you add the egg-lemon mixture.

Phase 3: Making the Avgolemono Sauce (The Crucial Step!)

8. Prepare Eggs and Lemon: In a medium-sized heatproof bowl, whisk the eggs vigorously until they are pale yellow and frothy, about 2-3 minutes. A hand mixer on low speed can also be used. Gradually whisk in the fresh lemon juice.
9. Temper the Eggs: This is the most critical step to prevent the eggs from scrambling. While continuously whisking the egg-lemon mixture, slowly ladle about 1 to 1.5 cups of the warm (not boiling!) broth from the soup pot into the egg-lemon mixture, one ladleful at a time. This gradually raises the temperature of the eggs. If you add the hot broth too quickly, the eggs will cook and curdle.
10. Combine with Soup: Slowly pour the tempered egg-lemon mixture back into the pot of soup, stirring gently but constantly as you pour.
11. Gently Heat (Do Not Boil!): Continue to cook the soup over very low heat, stirring gently and constantly, for another 3-5 minutes, or until the soup has thickened slightly and is heated through. Absolutely do not let the soup come to a simmer or boil once the avgolemono sauce has been added, as this will cause the eggs to curdle. The soup should be steaming and creamy.
12. Taste and Adjust: Taste the soup. Add more salt, freshly ground black pepper, or a bit more lemon juice if desired, to reach your preferred balance of flavors.
13. Serve: Ladle the Avgolemono into bowls immediately. Garnish with fresh chopped dill or parsley, and an extra grind of black pepper.