Preheat oven to 400F. Drizzle a little olive oil on the bottom of a square baking dish; spread it around and set aside.
In a large skillet, bring 1/2 inch of salted water to a boil. Add the broccoli and cauliflower; cover and cook over high heat until crisp-tender, about 4 minutes. Drain the vegetables in a colander and set aside.
Wipe out the skillet, add 2 tablespoons olive oil and heat over medium-high heat. Add onions and cook for 3 minutes, or until softened.
Stir in minced garlic and continue to stir and cook for 1 minute. Add broccoli and cauliflower to the skillet. Season with salt and pepper and continue to cook for about 2 minutes, or until lightly golden.
Remove from heat and transfer the vegetables to previously prepared baking dish; set aside.
Wipe the skillet and add 1 tablespoon olive oil, 3/4 cup shredded cheddar cheese, corn starch, evaporated milk, mustard, salt, pepper, and paprika.
Cook over low heat, stirring constantly with a whisk until melted and thickened; about 2 minutes.
Pour cheese sauce over broccoli and cauliflower; stir to combine. Sprinkle remaining cheeses over the top.
Bake for 20 minutes, or until lightly browned and bubbly. Remove from oven and serve immediately.