Cut a 2 1/2-inch hole from the center of one slice of the bread. Melt butter in a medium nonstick skillet over medium-low heat. When it foams, place the cut slice of bread and the round in skillet and toast lightly, about 1 minute.
Crack the egg into the hole and season with salt and pepper, and cook until golden brown, about 2 minutes.
Flip the egg bread, sprinkle half of the cheese on bread and top with ham. Spread mustard on second piece of bread and close sandwich. Cook until well browned and bottom part of cheese begins to melt, about 1 minute more. Flip whole sandwich and cook until all cheese is melted, about 2 to 3 minutes more. Serve.
Recipe courtesy of Aida Mollenkamp