Begin by blanching the broccoli, cauliflower, and carrots in boiling water for about 3-4 minutes until just tender but still bright in color. Drain the vegetables and transfer them to an ice bath to prevent further cooking.

In a medium saucepan, melt the butter over medium heat, then whisk in the flour to form a roux. Gradually add warm milk, stirring continuously until the mixture thickens.

Remove from heat and stir in the shredded cheddar and mozzarella, mixing until melted and smooth. Season with salt, pepper, garlic powder, and a pinch of nutmeg.

Lightly grease a baking dish, and layer the blanched vegetables evenly across the bottom. Pour the creamy cheese sauce over the vegetables, ensuring they are well-coated.

In a small bowl, toss together the breadcrumbs with melted butter and a sprinkle of cheese, creating a crunchy topping. Evenly spread this mixture over the casserole.

Preheat your oven to 375°F (190°C). Place the assembled casserole in the oven and bake uncovered for 25-30 minutes, or until the top is bubbly and golden brown.

Once baked, remove the casserole from the oven and let it cool for about 10 minutes before serving.