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.
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.
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.
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 stir it.