In a 6 to 8-quart stockpot, combine the chicken, water, and 2 tablespoons salt. Bring to a boil over medium-high heat; immediately reduce the heat to a very low simmer, and skim the foam from the surface. Add the leek, carrot, and bay leaves and continue to simmer with the chicken until the chicken is thoroughly cooked, about 45 minutes to 1 hour.
Meanwhile, heat the oil in a large skillet over medium-high heat. Add the onion and cook, stirring, until translucent, about 6 minutes. Remove from the heat and set aside.
Remove chicken from the broth, and allow meat to cool. Strain the broth and skim the fat. (Place the broth in the refrigerator to make it easier to skim.)
When the chicken is cool enough to handle, pull the meat from the bones and discard the skin. Dice the meat into large cubes; refrigerate until ready to use.
Return the broth to high heat, add the rice and onion and bring to a boil. Reduce the heat and simmer until the rice is almost cooked through, about 20 minutes. Add the chicken and reduce the broth to a low simmer.
In a medium sized bowl, beat the lemon juice, eggs, and pepper. Ladle 2 cups of hot broth into a measuring cup with a pourable spout. While whisking, slowly pour the 2 cups of broth into the egg mixture. Pour the egg mixture back into the pot with the remaining 1 tablespoon of salt. Stir well to blend. Divide among bowls and serve immediately.