Make the burgers, remove from the pan and place them on a plate in a warm oven. While the burgers are cooking, wash and thinly slice the mushrooms. When the burgers have been removed from the pan, add the mushrooms to the pan that the burgers were cooked in. Fry for 1 minute then stir in the horseradish sauce plus salt and freshly-ground pepper to taste. Spoon some of this sauce over each burger then sprinkle with grated cheese. Place under a preheated moderate grill (broiler) until the cheese has melted.