Prepare the Broccoli: Preheat the oven to 375°F (190°C). Bring a saucepan of salted water to a boil and add the broccoli florets. Cook for 2–3 minutes until bright green and slightly tender. Drain well and set aside.
Cook the Aromatics: Melt 2 tablespoons of butter in a medium saucepan over medium heat. Add the chopped onion and cook for 3–4 minutes until softened. Stir in the minced garlic and cook for another 30 seconds until fragrant.
Make the Creamy Sauce: Sprinkle the flour over the onion mixture and stir continuously for about 1 minute. Gradually whisk in the milk and vegetable broth, making sure no lumps form. Cook for 3–5 minutes until the sauce thickens slightly.
Add the Cheese: Reduce the heat to low. Stir in the sour cream, 1½ cups of shredded cheddar cheese, paprika, salt, and black pepper. Mix until the cheese melts and the sauce becomes smooth and creamy.
Combine the Rice and Broccoli: Place the cooked rice and drained broccoli in a large mixing bowl. Pour the cheese sauce over the mixture and fold gently until everything is evenly coated.
Transfer to the Baking Dish: Lightly grease a 9 × 13-inch baking dish. Spoon the broccoli and rice mixture into the dish and spread it into an even layer. Sprinkle the remaining ½ cup cheddar cheese over the top.
Prepare the Topping: Combine the breadcrumbs with 1 tablespoon of melted butter. Scatter the mixture evenly over the cheese layer to create a lightly crisp topping.
Bake Until Golden: Bake uncovered for 25–30 minutes, until the casserole is hot and bubbling around the edges and the breadcrumb topping is golden. Let it rest for 5–10 minutes before serving.