Preheat oven to 400 degrees F.

Bring a large pot of salted water to a boil over medium heat. Add the Brussels sprouts and cook for 5 to 7 minutes, until bright green and beginning to soften. Remove the sprouts with a slotted spoon and drop into a bowl of ice water to stop the cooking process.

Make the sauce by melting the butter and flour together in a saucepan over medium heat. Cook, stirring, until smooth and bubbling, about 1 minute; slowly whisk in milk and continue to cook, whisking frequently, until thick and creamy, 2 to 3 minutes. Season the sauce well with salt and heavily with pepper.

Halve the Brussels sprouts through the core and put them in an even layer in a 2 quart baking dish. Pour the sauce over the sprouts and sprinkle the cheese evenly over the top. Bake in the center of the oven for 10 to 15 minutes until the top is golden and bubbling.