Prepare chicken by removing excess fat. Fill a pot with enough water to cover chicken and bring to a boil. Boil for 5 minutes. Remove chicken and pour out water. Rinse pot.
Return chicken to pot and add enough water to cover chicken (at least 12 cups of water). Add carrots, celery, potato and leek. Bring to a boil, and then reduce heat to medium. Cook, covered, for approximately one hour. Check the pot approximately 10 – 15 minutes into the cooking and remove any froth that has come to the surface of the water.
Once the chicken has cooked remove it to a platter. Set aside.
Using a slotted spoon, remove the vegetables from the chicken broth. Set aside with the chicken.
Strain the broth into a large clean bowl using a fine sieve. Measure the liquid that you have. You will need 12 cups of liquid total. Add enough water in order to reach 12 cups.
Pour this stock into a pot. Rinse the arborio rice and then add it to the stock. Add salt.
At this point remove about 1 – 2 cups of the stock and keep it in a bowl, to cool slightly. Set it aside.
Cook the rice over medium high heat, stirring regularly (especially during the first 5 – 10 minutes to prevent the rice from sticking to the bottom of the pot) until the rice is cooked, about 20 minutes . If you find your soup is too thick, add some boiled water.
Remove pot from heat.
Separate eggs. Using a hand held mixer, or a fork, (or, if you have it, a rotary handheld egg beater!) beat egg whites until frothy (you will NOT create soft peaks…that’s okay). Once the egg whites are frothy and foamy, add the egg yolks. Continue beating. Then, add the lemon juice. Beat together until combined and then slowly stream in the cup of stock that you had set aside, and which should have cooled so that it is warm but not hot. Continue to beat the egg / lemon mixture until all of the stock has been incorporated.
Take this egg / lemon mixture and pour it slowly into the soup, stirring well. Give your pot of soup a few more good stirs after all the avgolemono has been added.
Serve in individual bowls. At this point you can choose to place cut up pieces of chicken directly into the servings of soup or do as our parents always did; serve the chicken, cut into pieces, on a large platter along with the boiled vegetables. Sprinkle lightly with salt. It is then up to each person to decide how much chicken (if any) to add to their bowl.