In a medium pot, bring the rice, broccoli, and 5 cups of water to a rolling boil for 2-3 minutes. Cover and let simmer on low for 20-25 minutes until rice is fully cooked. Turn off the burner, uncover.
Meanwhile, saute the garlic, onion, and mushrooms in a skillet and set aside. In the same pan, melt the butter on medium. Whisk in the flour and cook for 1-2 minutes. Turn the heat down for a moment and add the milk and broth. Add the garlic, onion, and mushrooms. Bring to a boil, reduce heat, and simmer for 10-15 minutes, stirring regularly, until it reaches the desired consistency (it will thicken some while cooling.)
Preheat the oven to 350 degrees. Combine the cream soup, rice, broccoli, and remaining ingredients in a large bowl, reserving .5 cups of cheese. Grease a 13x9 pan, then fill with mixture. Top with remaining cheese. Bake for 25-30 minutes until heated through.