Add the uncooked rice to the skillet and stir to coat in the butter and oil. Pour in the chicken broth and season with pepper. Stir to combine and bring to a boil.
Once boiling, reduce heat to low, cover, and simmer for 15 minutes.
Add the broccoli, cover, and cook for another 5 minutes to let it steam.
Remove the pot from heat and let the rice rest with the lid on for another 5-10 minutes.
Stir to mix the broccoli evenly with the rice.
Add the cooked chicken with plate juices, cream of chicken soup, milk, sour cream, and 1 cup of cheddar cheese. Stir to combine and taste. Adjust seasonings if needed.
Top with the remaining 1 cup of cheese, cover, and bake at 350°F for 15 minutes.
Optionally, uncover and bake for another 10 minutes or broil for a couple of minutes to achieve a browned cheesy top.
Serve warm.
Notes
Let the casserole sit for a few minutes after cooking to enhance the flavors.