Stock: Add all the ingredients to a large stock pot along with 1 gallon of cold water.
Put the stock pot onto the stove and cook over medium heat and simmer for about 1 hour or until the chicken is cooked through. You may need to go and skim some of the scum from the top of the stock.
Strain the stock through a fine-mesh strainer or chinois, discard the vegetables, set the chicken to the side, and put the stock back in the pot and back on the burner over low to medium heat.
Soup: Add in the onions, leeks, carrots, celery, and garlic to the stock and cook for about 5-7 minutes and then add in the rice and cook until the vegetables are tender, and the rice is cooked, about 5-10 additional minutes.
In the meantime, remove the meat from the cooked chicken and finely chop it. Add all the cooked chopped chicken or as little as half of it to the soup and turned the heat down to low. See note on leftover chicken.
In a bowl whisk together the egg yolks, egg, lemon zest, and lemon juice until combined and frothy.
Add about a ½ cup of chicken stock in the soup to the bowl with the egg yolks and lemon and continuously stir it while pouring in the stock to help temper the eggs.
Once combined, add egg mixture to the pot with the soup and gently mix it in using a spoon.
Season the soup with salt and pepper and serve it with optional chopped fresh parsley, or dill, or both.
Make-Ahead: You can make this recipe up to 1 day ahead of time. Keep it cold in the refrigerator before reheating it and serving it.How to Store: Cover and keep cool for up to 5 days in the refrigerator. Cover and freeze for up to 3 months. Thaw in the refrigerator for 1 day before reheating and serving.How to Reheat: Add the desired amount of avgolemono to a medium-size pot and heat over low heat until hot. Adjust the seasonings with salt and pepper and serve. You can use any sort of precooked or rotisserie chicken in this recipe and you will just need to use chicken stock instead of making it homemade as I did. Feel free to use whatever kind of rice you’d like; I’ve even seen orzo used in this recipe.You can also add things like turnips and potatoes to this recipe.The scum that collects on top of the stock is just impurities from the chicken and vegetables so it’s best to remove it using a spoon or ladle and discard it.You do not have to use all the cooked chicken in this recipe, feel free to put it in a plastic bag and freeze up to half of it.If your eggs are not getting frothy while whisking, add them to a blender or a food processor and blend or process for about 10-15 seconds.You can also cook the rice separately and simply add some of it to the soup when ready to serve it.