Prepare the broccoli and cauliflower florets by washing, drying, and chopping into bite-sized pieces.
Boil a small amount of water in a large pot and add the florets to a steamer basket over the water. Steam the broccoli and cauliflower florets just until bright in colour and only slightly tender (about 3 minutes). Set aside.
Preheat the oven to 375 degrees Fahrenheit and grease a large 9-inch by 13-inch (or equivalent) baking dish.
Heat a medium pot over medium-low heat and add the butter.
Melt the butter in the pot and add the flour, whisking constantly for about 1-2 minutes.
Slowly pour in the milk, whisking constantly to remove lumps.
Add the nutmeg and salt and pepper, whisking to combine.
Add the grated cheeses and stir gently until melted and the cheese sauce is smooth.
Add the steamed veggies to the bottom of the baking dish, pouring the hot cheese sauce over the top.
Stir just a little bit to be sure all the veggies are well coated in the sauce.
Mix the breadcrumbs and parsley in a small bowl together with the melted butter and sprinkle the mixture evenly over the top of the veggie and sauce.
Bake the casserole, uncovered, for 25-30 minutes or until the cheese sauce is bubbling and the topping is golden brown.