Assemble sandwich by first smearing mayonnaise on one side of one slice of bread. Spread 1/2 teaspoon of softened butter onto one side of the other slice of bread. Layer cheese, ham, and bacon onto the side of bread with butter spread onto it. Top with second slice of bread with mayonnaise side toward bacon.
Heat a cast iron skillet over medium-low heat and add a tablespoon of butter to the skillet to melt. Once butter has melted, add sandwich and toast until golden brown. Flip sandwich, adding additional butter if needed. Toast until second side of sandwich is golden brown. Remove from skillet and slice in half, diagonally.