Put the whole chicken and water in a large pot. Bring to a gentle simmer over medium heat, skim foam, then reduce heat to maintain a bare simmer and cook until the chicken is very tender, 45–60 minutes, until the meat pulls easily from the bone.
Lift the chicken to a platter. Strain the broth through a fine-mesh sieve back into the pot. Discard the skin and bones and reserve the meat for another use.
Return the broth to a simmer. Season with the kosher salt and black pepper, then stir in the white rice. Reduce heat to low and cook uncovered, stirring occasionally, until the rice is tender, 15–18 minutes.
In a large bowl, whisk the large eggs until light and frothy, about 1 minute. Slowly whisk in the fresh lemon juice.
Ladle about 2 cups of hot broth from the pot into the egg-lemon mixture in a thin stream, whisking constantly, until the mixture is warm to the touch.
Reduce the heat under the pot to very low. While stirring the pot, slowly pour the tempered egg-lemon mixture back into the soup. Cook gently without simmering, stirring constantly, until the soup turns silky and lightly thickened and just coats a spoon, 2–3 minutes.
Remove from heat. Taste and adjust seasoning to taste. Let rest 2 minutes, then ladle into bowls and serve immediately.