Sauté vegetables: In a large pot or Dutch oven, heat butter and olive oil over medium heat. Add celery, carrots, and onion. Cook until softened, about 5 minutes.
Add garlic and flour: Stir in garlic and cook for 1 minute. Sprinkle flour over vegetables and stir well, cooking for another minute to remove raw flour taste.
Add broth and seasonings: Slowly whisk in chicken broth. Stir in Italian seasoning, salt, and pepper.
Cook chicken: Add chicken breasts to the pot. Bring to a gentle boil, then reduce heat to simmer. Cover and cook for 20 minutes, until chicken is fully cooked.
Shred chicken: Remove chicken, shred with two forks, and return to the pot.
Cook orzo: Stir in uncooked orzo. Simmer for 10 minutes, or until pasta is tender.
Finish with lemon and parsley: Stir in lemon juice and fresh parsley. Taste and adjust seasoning as needed.
Serve: Ladle into bowls and serve with extra lemon wedges if desired.