Make the avgolemono: Place the egg whites in a metal bowl, if using a hand-held whisk, or in the bowl of a stand mixer outfitted with a whisk attachment. Whip at medium speed and then at a slightly higher speed, until a stiff meringue forms. In a separate, metal bowl, with a clean whisk, whisk the yolks until creamy and while whisking add the lemon juice in a slow, steady stream until thick and creamy. Using a rubber spatula, fold the meringue into the yolks thoroughly. Take a ladleful of the soup broth, careful not to catch any rice, trahana or pasta, and add it to the egg-lemon mixture in a very slow but steady stream, whisking vigorously all the while. Repeat with a second ladleful. Continue adding hot broth to the egg-lemon mixture until about half the pot juices are worked into the egg-lemon mixture, tempering it. Quickly pour this mixture into the pot, tilt so that it is evenly distributed, taste for salt and pepper or additional lemon juice, and serve, garnished with parsley sprigs.