Melt the butter in a saucepan, add the flour and stir until it forms a thick paste. Remove from the heat and gradually whisk in the milk and cream. Return to the heat and stir until the mixture has thickened.
Add the grated cheese, mustard, salt and freshly ground black pepper and simmer for five minutes over a low heat.
Place the quartered Brussels sprouts into a pan of salted boiling water and boil for 3-4 minutes, then drain.
Spoon the cooked Brussels sprouts into the base of a small gratin dish and pour over the cheese sauce. Place in the oven and bake for 8-10 minutes, or until golden-brown on top. Scatter with chopped fresh chives and serve.