In a large skillet, melt butter over medium heat. Sauté onion and broccoli until tender, about 5 minutes.
Sprinkle flour over veggies and stir to incorporate. Slowly whisk in milk and continue stirring to scrape up the brown bits stuck to the bottom of the pan.
Simmer slowly to thicken sauce, then season with salt, pepper and ground mustard. Melt cheese into sauce, then stir in cooked rice. Heat through 2-3 minutes, then serve.