Break up the chicken and shred the meat. Add to the soup. Add the sparkling or white wine and continue cooking another 6 to 7 minutes. Add half the lemon juice to the soup and stir.

Make the creamy 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 remaining lemon juice in a slow, steady stream until thick and creamy. Fold the meringue into the yolks thoroughly. Take a ladleful of the soup broth, careful not to catch any rice, 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.