Place the chicken in a large pot with 2 quarts cold water, enough to cover the chicken. Bring to a boil and reduce heat to low, skimming when necessary.
In a separate pan, heat 2 tablespoons of olive oil and add the onions. Sweat the onions until clear. Set aside.
When chicken is cooked through, remove from the broth. Let the chicken cool and pull the meat from the bones. Dice into large cubes. Set aside.
Add the onion, bay leaves, leek, and carrot to the broth and simmer for 1 hour. When finished, remove the carrot and leek from the broth and add the rice. Bring to a boil and then turn heat to medium high to simmer until the rice is cooked to al dente, about 30 minutes. Add the chicken back into the broth. Add more water if needed.
In a small bowl beat the eggs and lemon juice together. Pour 2 cups of broth slowly into the bowl of egg and lemon, whisking continuously. Once all the broth is incorporated, add the mixture into the pot of chicken soup and stir to blend well throughout. Season with salt and pepper. Serve hot.