Preheat the oven to 375 degrees F (190 degrees C).

Bring a large pot of lightly salted water to a boil. Add broccoli and cauliflower; cook, uncovered, until tender yet firm to the bite, 2 to 3 minutes. Drain in a colander and immediately immerse in ice water for several minutes to stop the cooking process. Drain and transfer to a baking dish.

While the broccoli and cauliflower are cooking, melt butter in a saucepan over medium heat. Add flour and stir until browned and bubbling, 3 to 4 minutes. Whisk in half-and-half and mustard; bring to a boil. Reduce the heat to medium-low and simmer until sauce thickens, about 2 minutes.

Stir 2/3 cup Cheddar cheese, Parmesan cheese, and paprika into sauce; cook and stir until both cheeses are melted. Season with salt and pepper.

Pour sauce over broccoli and cauliflower in the baking dish; sprinkle 3 tablespoons Cheddar cheese and bread crumbs over top.

Bake in the preheated oven until topping is lightly browned, 15 to 20 minutes.