In a 6 to 8-quart stockpot, combine the chicken, water, and 2 tablespoons salt. Bring to a boil over medium-high heat. 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. The internal temperature of the chicken needs to be 165F.
Meanwhile, heat the oil in a large skillet over medium-high heat. Saute the onion until translucent, about 6 minutes. Remove from the heat and set aside.
Remove chicken from the broth and allow meat to cool before dicing or shredding. Strain the broth (optional) and skim the fat.
Return the broth to the stove, bring it back to a boil, and add the onion and rice or orzo. Reduce the heat and simmer until the rice/orzo is almost cooked through, about 20 minutes for rice, 6-8 for orzo.
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 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