Preheat your oven to 350°F (175°C) and lightly grease a 9x13 inch casserole dish.
If using raw chicken, season with salt, pepper, and Italian seasoning. Heat butter over medium heat, add chicken, and cook until golden brown on all sides, about 5 minutes. Set aside.
In the same pot, add chicken broth, rice, and olive oil. Bring to a boil, then reduce heat to low, cover, and simmer for 7 minutes.
Add broccoli florets on top of the rice without stirring, cover, and cook for an additional 8 minutes.
In a large mixing bowl, combine cooked chicken, cream of chicken soup, milk, sour cream, half of the cheddar cheese, garlic powder, and thyme (if using). Mix until smooth.
After the rice and broccoli are done, remove from heat and let sit, covered, for 10 minutes. Add the rice and broccoli to the creamy mixture and stir until evenly combined.
Spread the mixture evenly in the prepared casserole dish and top with the remaining cheddar cheese.
In a small bowl, combine crushed Ritz crackers with melted butter and sprinkle over the casserole.
Cover with aluminum foil and bake in the preheated oven for 15 minutes. Remove the foil and bake for an additional 10-15 minutes until the cheese is melted and the topping is golden brown.
Let the casserole sit for a few minutes before serving to allow the flavors to meld. Enjoy!