In a large stockpot, bring the chicken stock to a boil. 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 to a bowl. Once cooled, shred the chicken and reserve.
Add the orzo or rice to the pot with the chicken stock and cook until al dente, 5 to 8 minutes.
In a large bowl, whisk the egg whites to soft peaks. In a medium bowl, whisk together the egg yolks and most of the lemon juice, reserving some 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, whisking constantly and adding the hot liquid slowly. Very slowly, add the warm whipped egg/lemon mixture into the pot, whisking constantly to prevent any curds or clumps from forming. Continue to cook over medium-low to medium heat, whisking, until the soup thickens, about 10 minutes more.
Add the shredded chicken and any reserved juices to the soup. Add the lemon zest and season with salt and plenty of freshly cracked black pepper. Taste and add additional lemon juice if needed. Garnish with chopped parsley.