Preheat oven to 400°F. Prepare an 8x11 casserole dish by greasing lightly with olive oil or butter.
In a large skillet, over medium-high heat add 1 tablespoon of cooking oil and sear the chicken for about 5-6 minutes. Set aside.
In the same skillet, add the remaining cooking oil. And once hot, saute the onion until it starts to get brown.
Add in the flour and whisk quickly for about a minute or so.
In batches, stir in milk, continuously whisking. When all the milk is added, add in the spices and season with salt and black pepper.
Simmer the mixture on low heat until it thickens. Stir every once in a while. Remove from the heat and add 1-½ cup of cheddar cheese. Stir until cheese is melted and incorporated into the mixture.
In a large bowl or right in the prepared casserole dish, mix rice, broccoli, chicken, and sauce.
Sprinkle the remaining ½ cups cheese on top. Bake for 20-25 minutes.
Serve while the casserole is still warm and the cheese melty.