Preheat oven to 350° F. Lightly grease a 9x13-inch baking dish.
Bring a large pot of water to a boil. Add broccoli and boil for 3-4 minutes, or until crisp-tender (it will cook more in the oven). Transfer broccoli to a colander, rinse with cold water, and drain very well. Set aside.
In a medium saucepan, melt the butter over medium heat. Add the onion and cook, stirring often, until softened, about 4 minutes. Whisk in the minced garlic and flour and cook, stirring, for 30 seconds.
Whisk in the milk and broth, followed by the oregano, garlic powder, salt and pepper. Cook, stirring occasionally, until the mixture starts to bubble and thickens, about 3-4 minutes. Remove pan from the heat and stir in 1 ¼ cups of the shredded cheddar cheese, a little at a time, and the Parmesan cheese until the cheese melts into the sauce.
In a large bowl, combine the broccoli and cooked rice. Pour the cheese sauce over and stir gently to combine.
Transfer the mixture to the baking dish. Sprinkle the remaining shredded cheddar over the top. Bake for 30-35 minutes, or until hot and bubbly and cheese is lightly browned on top. Let rest for 5 minutes before serving.