Add the chicken to a stockpot and cover with 3 quarts of cold water. Set the pot over high heat and bring the water to a steady simmer. Using a small sieve or ladle, scoop off any foam that rises to the surface, around 10 minutes. Lower the heat to a gentle simmer, cover with a lid and cook until the chicken is easily falling off the bone, around 45 minutes. Reserve the broth and transfer the chicken to a plate. Let the meat cool then shred it with your hands or two forks. Discard the bones and set the meat aside.

To the broth, add the carrot, celery, onion, garlic, rice, bay leaves and 1 tablespoon kosher salt. Raise the heat to a steady simmer and cook, stirring occasionally, until the vegetables are tender and the rice is cooked, around 15 minutes.

Meanwhile, separate the egg yolks and whites into two different bowls. Whisk the egg whites until frothy. Add the yolks to the whites and whisk to combine. Add the lemon zest and lemon juice and whisk to combine. While continuously whisking, slowly ladle in 1.5 cups of the hot chicken broth into the egg mixture.

Remove the pot of soup from the heat. While continuously whisking, slowly pour the tempered egg mixture back into the soup. Return the shredded chicken to the soup and stir through to combine. Taste the soup for seasoning and adjust with salt and lemon juice as desired.