Preheat oven to 400 F. Spray a 9x9 inch baking dish with cooking spray.
Thinly shave brussels sprouts in a food processor.
Melt butter in a small saucepan. Add shallot and garlic and cook for 2 minutes, stirring to prevent burning. Add flour and cook for 30 seconds. Slowly pour the half & half. Add salt, pepper and nutmeg. Bring to a boil, stirring frequently. Remove from the heat immediately.
Pour mixture over the brussels sprouts, top with cheese. Cover with aluminum foil and bake for 20 minutes.
Remove the foil and cook for 5 more minutes. Check if the Brussels Sprouts are fully cooked and tender. Bake for additional 5 minutes, if necessary, until the half & half is absorbed and the top is golden.