1. Cook gnocchi according to package directions and drain. Transfer to a 9×13-inch (23×33-centimeter) pan.
2. Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius).
3. In the same pot, heat milk and flour, stirring until thickened. Add heavy cream, salt, and pepper, and stir. Mix in cheddar and mozzarella cheeses and stir until melted.
4. Pour the mixture over the gnocchi.
5. Add broccoli to one half and sautéed mushrooms to the other half and lightly mix each side so the vegetables are incorporated. Top the broccoli half with a light sprinkle of panko crumbs and the mushroom half with cornflake crumbs. Sprinkle Parmesan over the whole thing.
6. Bake until the cheese on top is brown and the whole thing is bubbly, about 20 minutes.