Cover chicken breasts with at least 1 inch of water. Add 1 teaspoon salt and bring to a gentle boil, skimming off the foam. Reduce heat and cover, simmering for ten minutes. Turn off the heat.
Let the chicken sit in the cooking liquid, covered, for another 5 to 10 minutes, then remove the breasts and cut into bite size pieces. Place the chicken pieces in a bowl with enough of the cooking liquid to cover, and put aside. (This keeps the chicken moist while the soup is assembled.)
Place the chicken broth in a large pot, and bring it to a boil over high heat. Add the rice, and cover the pot. Decrease the heat to low, and simmer for 15 minutes. Turn off heat.
In a medium bowl, whisk the lemon juice into the egg yolks.
Slowly whisk a ladleful of hot broth (or approximately 1/4 cup) into the lemon/egg yolk mixture. Add several more ladlefuls of broth, continuing to whisk so that the egg yolks do not curdle.
Whisk the lemon/egg yolk mixture into the large pot of broth and rice. Drain the chicken pieces, and add them to the soup. Turn the heat to low, and do not let the soup boil. Stir occasionally for approximately 5 minutes. Turn off the heat.
If serving the soup cold, cover and place the pot in the refrigerator, when the soup has cooled slightly. Cool to desired temperature.