Preheat your oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish with the butter and set it aside.
Place a large skillet over medium-high heat and add the olive oil. When the oil is hot, add the ground beef and cook, breaking it up with a spoon, until browned and no longer pink, about 6–8 minutes.
Add the chopped yellow onion to the skillet with the beef and cook, stirring occasionally, until the onion is softened and translucent, about 3–4 minutes.
Stir in the minced garlic and cook for 30 seconds to 1 minute, just until fragrant, being careful not to let it burn.
Add the broccoli florets to the skillet and stir to combine with the beef mixture. Cook for 3–4 minutes, just to soften the broccoli slightly while keeping some crunch.
Turn off the heat under the skillet. Add the cooked white rice, salt, black pepper, dried thyme, and paprika to the beef and broccoli mixture. Stir until everything is evenly combined.
In a large mixing bowl, whisk together the condensed cream of mushroom soup and the milk until smooth and well blended.
Add 1 cup of the shredded cheddar cheese to the soup and milk mixture and stir until the cheese is evenly distributed.
Transfer the beef, broccoli, and rice mixture from the skillet into the mixing bowl with the creamy cheese sauce. Stir thoroughly until all ingredients are well coated.
Spoon the combined mixture into the prepared baking dish, spreading it out into an even layer with the back of a spoon or spatula.
Sprinkle the remaining 1/2 cup of shredded cheddar cheese evenly over the top of the casserole, followed by the grated Parmesan cheese.
Cover the baking dish loosely with foil and bake in the preheated oven for 20 minutes to allow the casserole to heat through.
After 20 minutes, remove the foil and continue baking for another 10–15 minutes, or until the casserole is bubbling around the edges and the cheese on top is melted and lightly golden.
Carefully remove the casserole from the oven and let it rest for about 5–10 minutes to set slightly before scooping and serving.