Bring 4 cups salted water to a boil, add brussels sprouts and cook 10 minutes. Drain and refresh under cold water. Drain again.

Meanwhile, melt butter in a 3-quart saucepan. Add shallots and cook over medium heat, stirring, until light brown. Pour contents of pan through a fine strainer into a dish, pressing to remove as much butter as possible from shallots. Place shallots on paper towel to drain. Return butter to saucepan.

If serving immediately, preheat oven to 400 degrees. Place saucepan over medium heat and cook until butter has a nutty aroma and is turning brown. Whisk in flour and cook until mixture is light brown. Whisk in stock and cook until sauce has thickened. Add lemon juice, salt and pepper to taste, and nutmeg. Add chestnuts and brussels sprouts, folding ingredients together.

Transfer to an 8-cup baking dish. Scatter shallots on top. Bake about 15 minutes. Serve.