Preheat the oven to 400 degrees F. Put the flour and broth into a large saucepan and whisk until the flour is dissolved. Place the saucepan over medium heat and whisk continuously until the mixture comes to a boil; continue to whisk until the mixture is thick enough to coat the back of a spoon, 10 to 12 minutes. Remove the saucepan from the heat and whisk in 1 1/2 teaspoons salt and 1/2 teaspoon pepper. Add the Cheddar and whisk until melted. Add the broccoli florets and stir to coat.
Spray a 3-quart flameproof casserole dish with cooking spray. Toss the brown rice and chicken in a large bowl. Pour the broccoli-cheese sauce on top and mix well. Spread the mixture in the casserole dish and top with the Parmesan. Cover with foil and bake until bubbling and hot, about 25 minutes. Remove the foil and broil until the top is nicely browned, 2 to 5 minutes Serve hot.
Make-Ahead Tip: Freeze the baked casserole for up to 2 weeks. Cover in foil and reheat at 350 degrees F until hot, 35 to 45 minutes.
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