Use a paring knife to cut an X in the flat side of each chestnut shell. Toss into a pot of boiling water and cook for 10 minutes. Peel away the outer shell and brown skin. Return the peeled chestnuts to the pot of water and boil until tender, about 25 minutes.
Blanch the Brussels sprouts in boiling salted water for about 8 minutes or until they are just tender. Take care not to over cook. Drain the Brussels sprouts well and cut in half lengthwise. Melt butter in a skillet and saute the shallots. Add the chestnuts and Brussels sprouts, toss to coat with the butter. Season with salt, pepper and nutmeg. Cook for 5 minutes until heated through