In a medium bowl, mix the grated cheese, mayonnaise, Dijon mustard, and horseradish. Place a liberal layer of the cheese on 1 side of each of the 6 slices of bread. Place the ham and tomato slices on 3 of the pieces of the cheese topped bread, then top them with the remaining 3 slices of bread to form a sandwich. Place butter in a large saute pan over low to medium heat. Add the sandwiches to the melted butter and slowly brown them on both sides while allowing the cheese to melt inside. Cut the crusts from the sandwiches and then cut them crosswise into 8 hors d'oeuvre size portions. Serve warm.