Preheat the oven to 350 degrees F. Grease a 2-quart casserole dish.
Cut fresh broccoli florets into 1-inch pieces. Place the florets in a large microwave-safe bowl and add 3 tablespoons of water Cover and microwave for 2 minutes. Drain thoroughly.
In a large pot, melt the butter over medium heat. Add the onion and saute until translucent. Add the minced garlic and cook another minute.
Whisk in the flour, dry mustard, salt, and pepper. Cook another 1 to 2 minutes.
While whisking constantly, gradually pour in the milk. Continue whisking until thick and bubbly. It should be the consistency of cream soup.
Remove from the heat add the cream cheese and 1 1/2 cups of shredded sharp cheddar cheese. Stir until the cheese is melted.
Taste for seasoning. Add more salt and pepper if desired.
In the same large bowl stir the broccoli, 2 cups cooked rice, and cheese sauce together until completely combined. Pour into the prepared baking dish.
Sprinkle the remaining 1 cup of cheese over the top evenly.
Bake for 30-35 minutes, or until bubbly and golden brown.
Let rest for about 10 minutes before serving.