Preheat the oven to 350°F and grease a 9-by-13-inch casserole dish with butter or cooking spray.

In a large skillet over medium heat, melt the butter with the olive oil, then add the onion and garlic and sauté for 3 to 4 minutes, until the onion is translucent, softened, and fragrant but not browned.

Add the broccoli florets to the skillet and cook for another 5 minutes, stirring occasionally, until bright green and beginning to soften around the edges.

In a large mixing bowl, combine the cooked white rice, sautéed onion, garlic, and broccoli, 1 cup of the shredded cheddar cheese, milk, sour cream, salt, pepper, and paprika, stirring until the rice is coated and the sour cream is worked through the mixture.

Spoon the mixture into the greased casserole dish and spread it out evenly with the back of a spoon without pressing it down firmly.

Sprinkle the remaining 1/2 cup cheddar cheese over the top, followed by the grated Parmesan cheese.

Bake for 25 to 30 minutes, until the casserole is hot and bubbling and the cheese on top is golden brown, checking at 25 minutes.

Let the casserole sit for a few minutes after removing it from the oven, then sprinkle with chopped fresh parsley just before serving.