Preheat the oven to 375 degrees and lightly grease a medium baking dish.
Cook the rice according to package directions until tender, then set it aside.
Steam or blanch the broccoli florets for 3 minutes until bright green and slightly tender, then drain well.
In a saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1 minute.
Slowly pour in the milk while whisking, then add the salt and pepper. Cook until the sauce thickens.
Remove the sauce from heat and stir in 1 1/2 cups of the cheddar until melted.
Combine the cooked rice, broccoli, and cheese sauce in a large bowl, then transfer to the baking dish.
Sprinkle the remaining 1/2 cup cheddar over the top and bake for 20 minutes until the cheese is bubbly and lightly browned.