Preheat your oven to 375°F (190°C) and lightly grease a 9x13 inch baking dish.
In a large skillet, melt the butter over medium heat. Add the chopped onion and cook until translucent, about 4 minutes. Add the minced garlic and cook for 1 more minute.
Sprinkle the flour over the onions and garlic, stirring constantly for 1 minute to cook the raw flour taste.
Gradually whisk in the chicken broth and milk until the mixture is smooth. Bring to a gentle simmer and cook until slightly thickened, about 3-5 minutes.
Remove the skillet from the heat. Stir in 1.5 cups of the grated cheddar cheese, salt, and black pepper until the cheese is completely melted and the sauce is smooth.
In a large mixing bowl, combine the cooked diced chicken, cooked rice, and broccoli florets. Pour the cheese sauce over the mixture and gently toss to coat everything evenly.
Transfer the mixture to the prepared baking dish and spread it out into an even layer. Top with the remaining 1/2 cup of cheddar cheese.
Bake uncovered for 25-30 minutes, or until the casserole is bubbly and the cheese is lightly golden on top. Let rest for 5 minutes before serving.