Preheat the oven to 350° F. Cut chicken into bite-sized pieces and season with Italian seasoning and desired amount of salt and pepper.
Cook chicken in a large pot with melted butter until golden brown, then set aside. (Optional if using rotisserie chicken.)
Add chicken broth, olive oil, and rice to the pot. Bring to a boil, reduce heat, cover, and simmer for 6 minutes. Add broccoli, cover, and cook for 9-14 minutes until rice is cooked and liquid is absorbed.
Remove pot from heat and let stand, covered, for 10 minutes. Stir in cooked chicken, cream of chicken soup, milk, sour cream, thyme, garlic powder, and half the cheddar cheese.
Transfer mixture to a greased 9x13 casserole dish, top with remaining cheese, and bake covered for 15 minutes.
Notes
For a crisper top, uncover the casserole for the last 5 minutes of baking.