In a small saucepan, melt the butter, add the grated cheese, mustard, salt and pepper to taste.
Constantly stirring, heat the mixture over low heat without bringing to a boil. As soon as the cheese melts, remove from the heat.
Beat the egg with a whisk. Add to the cheese mixture and stir. Put again on a small fire. Cook, stirring actively, for 1 minute until smooth. Remove from heat.
Apply the filling to the bread slices.
Bake the sandwiches at 180 degrees for about 7-8 minutes (until golden brown).
Serve both hot and cold.