Begin by heating the olive oil in a large pot over medium heat. Once hot, add the diced onion, celery, and carrots. Sauté the vegetables for about 5 minutes until they are softened and fragrant.
Next, take the boneless chicken breasts and add them to the pot along with the chicken broth. Increase the heat to bring the mixture to a gentle simmer. Cover and cook for about 20 minutes, or until the chicken is completely cooked through and no longer pink in the center.
Carefully remove the chicken from the pot and place it on a cutting board to cool for a few minutes. While the chicken is cooling, stir in the orzo pasta and dried oregano into the simmering broth. Cook the orzo according to the package instructions, which typically takes about 8-10 minutes, until al dente.
As the orzo cooks, shred the slightly cooled chicken using two forks. Once shredded, return the chicken to the pot, mixing it back into the soup.
In a separate bowl, whisk together the eggs, fresh lemon juice, and lemon zest until the mixture is smooth and well combined.
To incorporate the egg mixture without curdling, slowly ladle a small amount of the hot soup broth into the egg mixture while whisking constantly. This process helps to temper the eggs. After combining, gradually pour the tempered egg mixture back into the pot while stirring gently.
Stir the soup thoroughly and season with salt and pepper to your liking. Allow it to simmer for an additional 2-3 minutes to ensure everything is heated through and well blended.
Serve the soup hot in bowls, garnished with freshly chopped parsley for a vibrant finish.