Melt 1 1/2 tablespoons butter in a nonstick skillet over medium-low heat. Place bread slices in the skillet on top of the melted butter.
Spread about 1/4 cup Cheddar cheese on one slice of bread; place the other slice of bread, butter-side up, on top of the cheese. Spread about 2 tablespoons of cheese on top of the sandwich.
Melt remaining 1/2 tablespoon butter in the skillet next to the sandwich. Flip the sandwich onto the melted butter so that the cheese-side is facing down. Spread remaining cheese on top of the sandwich. Cook sandwich until cheese on the bottom is crispy and caramelized, 3 to 4 minutes. Flip sandwich and cook until cheese is crispy and caramelized on the other side, another 3 to 4 minutes.