Preheat oven to 350 degrees. Slice 6 (1/2-inch) thick pieces out of the brioche loaf. Top 3 of the slices with thinly ham and the grated cheese. Place on a baking sheet and cook for approximately 5 minutes or until the cheese is melted and golden brown. Take a glass and cut out the middle of the other 3 slices of brioche. Heat the butter in a medium saute pan until hot and bubbly. Add the hollowed out slices and crack an egg in the middle of each piece. After 1 minute flip the slice over and cook 1 minute. Remove from the pan and place together with the ham and cheese, creating a sandwich. Serve hot.
Recipe courtesy of John Hogan, Savarin Restaurant, New York City