Sauté Aromatics and Chicken: In a large pot or Dutch oven, heat the olive oil over medium heat. Add the chopped chicken pieces and cook until lightly browned on all sides, about 5-7 minutes. Remove the chicken from the pot and set aside. Add the chopped onion, carrots, and celery to the pot. Sauté until softened, about 8-10 minutes. Add the minced garlic and cook for another minute until fragrant.
Simmer Broth and Orzo: Pour in the chicken broth and bring it to a boil. Add the orzo pasta, stirring to prevent sticking, and reduce the heat to a simmer. Cook the orzo according to package directions, typically 8-10 minutes, or until al dente. While the orzo cooks, shred or dice the cooked chicken into smaller, bite-sized pieces.
Prepare Avgolemono: In a medium bowl, whisk together the fresh lemon juice and eggs until well combined and frothy. This is your avgolemono mixture. To temper the eggs, slowly ladle about 1 cup of the hot broth from the simmering soup into the egg-lemon mixture, whisking continuously to prevent the eggs from scrambling. This step is crucial for a smooth soup.
Combine and Finish: Gradually pour the tempered egg-lemon mixture back into the main pot of soup, stirring constantly. Add the cooked chicken back to the pot. Heat gently for a few minutes, stirring frequently, but do not bring the soup to a boil after adding the egg mixture, as it can curdle. Season with salt and freshly ground black pepper to taste. Stir in the fresh dill just before serving for a burst of fresh flavor.
Serve: Ladle the warm Lemon Orzo Chicken Soup into bowls. Garnish with additional fresh dill or a lemon slice, if desired. Serve immediately and enjoy your hearty, flavorful meal.