Preheat the oven to 375°F and lightly grease a 9-by-13-inch baking dish. Season the chicken with salt, pepper, and paprika.
Heat the olive oil in a skillet over medium-high heat. Brown the chicken for 3 to 4 minutes, then transfer it to a plate.
Melt the butter in the skillet. Cook the onion until soft, then add garlic, flour, thyme, paprika, salt, and pepper.
Gradually whisk in the chicken broth and milk. Simmer for 2 to 3 minutes until the sauce is smooth and slightly thickened.
Add the rinsed rice, broccoli, chicken, and sauce to the baking dish. Stir gently to combine.
Cover tightly with foil and bake for 45 minutes. Remove the foil, top with cheddar cheese, and bake for 12 to 15 minutes more.
Rest the casserole for 10 minutes before serving.