Move a rack to one of the top oven shelves and preheat the broiler to low.
Over medium-high heat, melt the unsalted butter in a medium, oven-safe skillet. When it begins to sizzle, add the bread and allow it to toast on one side, approximately 2 minutes. Once golden, flip the bread and continue browning on the other side. Remove from the heat and top evenly with the cheese.
Carefully place the skillet under the broiler until the cheese has melted, approximately 20-30 seconds (keep an eye on it). Set aside.
Preheat a small or medium cast iron or heavy-bottom skillet over high heat. Crack the egg into a ramekin or small bowl. Spread some paper towels on a small plate and place nearby.
Once the pan is smoking hot, add the clarified butter or oil, and then carefully add the egg (I recommend wearing oven mitts during this process as the butter might splatter). The egg will immediately puff up and bubble. Holding the skillet handle and tilting the pan slightly away from yourself to avoid splatter, use a large, heat-resistant spoon and repeatedly ladle the butter over the egg to help the white set. This should not take longer than 60 seconds total. The bottom of the egg should be caramelized and crisp. Remove the pan from the heat and transfer the egg to the paper towels to briefly drain some of the excess oil.
Place the fried egg on top of the prepared bread and cheese (the heat from the egg will remelt the cheese if it has started to cool down).
Sprinkle with salt and pepper to taste. Serve immediately.