Heat the olive oil and butter in a 6-quart stock pot or Dutch oven over medium-high heat. Sauté the onion for 3 to 4 minutes, stirring occasionally. Add the rice, stir well, and cook for another minute or two. Add the chicken stock and submerge the chicken breasts in the liquid. Season the stock with salt and pepper. Cover and simmer over low heat for 20 minutes.

Remove the lid and skim any white foam off the top of the stock. Transfer the chicken to a cutting board and allow to cool slightly.

Whisk the eggs in a bowl until frothy. Whisk the lemon juice into the eggs and then, whisking constantly, add a ladle of the hot soup to temper the eggs. Then whisk the egg mixture back into the pot. Stir over low heat until the soup thickens.

Shred the chicken with two forks or cut it into bite-sized pieces. Stir the chicken pieces back into the soup with half of the lemon zest and parsley. Season with salt, freshly ground black pepper and more lemon juice to taste. Garnish with chopped fresh parsley and more lemon zest.