Sauté vegetables: In a large soup pot, heat the butter and olive oil over medium-high heat. Add chopped celery, carrots, and onion. Sauté for 5 to 7 minutes, or until vegetables are softened.
Add garlic and flour: Stir in the minced garlic and cook for 30 seconds until fragrant. Sprinkle in the flour and stir constantly for another minute to cook off the raw flavor.
Add broth and chicken: Slowly pour in the chicken broth, stirring continuously to blend with the flour. Add Italian seasoning and nestle the chicken breasts into the broth.
Simmer: Bring the soup to a boil, then reduce the heat and simmer for 15 minutes with the lid slightly ajar.
Add orzo: Stir in the orzo and continue to simmer uncovered for about 10 minutes, stirring occasionally to prevent sticking.
Shred chicken and finish: Remove the cooked chicken breasts, chop into bite-sized pieces, and return to the pot. Stir in lemon juice and chopped parsley. Season with salt and pepper to taste.
Serve: Serve hot, garnished with additional lemon slices or parsley if desired.