Place the cut chicken in a pot, add the broth, and cook to a boil. Once boiling, simmer until the chicken cooks, about 35 minutes.
Remove the chicken leaving the soup, and set aside. Skim any foam or debris, leaving a clear and flavorful broth.
Wash the rice, drain and then add it into the broth, and cook until tender for about 20 minutes.
Meanwhile, as the rice cooks, shred the chicken into bite-size pieces and put them back into the rice pot once the rice cooks. Ensure to maintain low heat at this point to avoid overcooking.
Break the eggs into a small bowl, mix with the lemon juice, and whisk thoroughly to combine well.
Scoop some broth from the pot and add to the egg-lemon mixture while constantly stirring to avoid curdling. This process is called tempering and hinders the egg from curdling when added to the soup.
Now add the egg mixture to the soup, little by little as you stir.
Simmer the soup for 5-8 minutes while occasionally stirring to achieve a thick, creamy texture.
Season your traditional avgolemono soup with pepper and salt to your liking. Stir in more lemon juice for an extra tangy flavor, and remove the soup from the heat.
Serve the soup with a garnish of parsley.