Steam broccoli florets until just tender, about 4 minutes, then drain and set aside for later.
Melt butter with olive oil in a large skillet over medium heat until hot and ready to use.
Add diced onion and minced garlic to the skillet, cooking until softened and smells good, about 3 minutes.
Stir in uncooked white rice, toasting gently until grains are coated and slightly golden looking.
Pour in chicken stock and half and half, add salt, bring to a boil, then turn down the heat.
Cover the skillet and cook for 15 minutes until the rice soaks up all the liquid and becomes tender.
Take off heat and stir in steamed broccoli, sharp cheddar cheese, and Parmesan until cheese melts.
Put mixture into a greased baking dish, spreading evenly with a spoon for uniform cooking and perfect texture.
Mix melted butter, Panko breadcrumbs, smoked paprika, and Parmesan cheese in a small bowl until well combined.
Sprinkle breadcrumb topping evenly over the casserole surface, covering completely for a golden, crispy finish after baking.
Bake at 375 degrees for 20 minutes until the topping is golden brown and crunchy all over.
Let the casserole rest for 5 minutes after baking to let flavors settle before serving to everyone.