Bring water to a boil in a saucepan. Add the broccoli florets and blanch for 2 minutes. Transfer the broccoli to a bowl filled with cold water and let it sit for 10 minutes, or until completely cooled. Drain and set aside.
Preheat the oven to 450°F (230°C). In a saucepan, melt half of the butter over medium heat. Whisk in the all-purpose flour, then stir in the milk. Season with salt, nutmeg, and ground black pepper. Stir constantly for 1 to 2 minutes, until the ingredients are fully combined. Turn off the heat.
In a butter-greased baking dish, arrange the broccoli florets in a single layer. Evenly sprinkle the grated sharp cheddar on top, then pour the milk mixture over the broccoli.
Melt the remaining half of the butter in a skillet over medium heat. Add the breadcrumbs and cook, stirring frequently, for 2 minutes, or until the breadcrumbs are golden brown. Turn off the heat and sprinkle the toasted breadcrumbs over the gratin.
Bake the broccoli gratin for 20 minutes, or until it is golden brown and bubbly. Serve immediately.