Bring the chicken stock to a boil in a large stockpot over high heat. Add the chicken breasts and then lower to a simmer. Simmer until the chicken is just cooked through and the internal temperature registers 160 degrees F, 15 to 20 minutes. Remove the chicken to a bowl. Once cooled, shred the chicken and reserve.
Add the rice to the pot with the chicken stock and cook until al dente, about 8 minutes.
Whisk the egg whites to soft peaks in a large bowl. Whisk together the egg yolks and most of the lemon juice in a medium bowl, reserving some juice to add later as needed. Fold the egg yolk mixture into the egg whites. Once combined, ladle in some hot stock to temper the egg mixture, stirring constantly and adding the hot liquid slowly. Add the lemon zest and shredded chicken with any reserved juices to the soup and, very slowly, add the warm whipped egg/lemon mixture. Whisk to incorporate the egg mixture and prevent any curds or clumps from forming. Continue to cook over medium-low to medium heat, whisking occasionally, until the soup thickens, about 10 minutes more.
Season with salt and plenty of freshly cracked black pepper and add the remaining lemon juice if necessary. Add most of the parsley. Garnish individual bowls with the remaining parsley.