Sauté the Aromatics: In a large, heavy-bottomed pot or Dutch oven, warm the olive oil over medium heat. Add the diced carrots, celery, and green onions. Stir frequently, cooking until the vegetables soften and become fragrant, about 4–5 minutes. Add the minced garlic and sauté for another 30 seconds, being careful not to let it burn.
Simmer the Broth and Rice: Pour in the chicken broth and add the bay leaves. Increase the heat to bring the liquid to a vigorous boil. Once boiling, stir in the rinsed rice along with a pinch of salt and freshly ground black pepper. Reduce the heat to low and let the soup simmer gently, uncovered, for about 20 minutes or until the rice is tender and cooked through.
Incorporate the Chicken: Add the shredded cooked chicken to the pot, stirring to combine. Allow the soup to continue warming on low heat as you prepare the avgolemono sauce.
Prepare the Avgolemono Sauce: In a medium-sized mixing bowl, whisk together the eggs and fresh lemon juice until smooth and slightly frothy. To temper the eggs and prevent curdling, slowly drizzle in two ladlefuls of hot soup broth into the egg-lemon mixture while continuously whisking. This gradual warming is essential to create a creamy sauce.
Finish the Soup with Sauce: Slowly pour the tempered egg-lemon mixture back into the pot with the soup, stirring gently but continuously. Immediately remove the pot from the heat to avoid cooking the eggs too quickly, which can cause the soup to separate. The broth should become silky and slightly thickened.
Garnish and Serve: Discard the bay leaves. Ladle the soup into bowls, garnish with freshly chopped parsley if desired, and serve immediately. Pair with crusty bread for a complete meal.