In a small bowl, combine the butter, parmesan cheese, and garlic powder. Set aside.
Place the 2 slices of bread on a cutting board; spread 1 side of each slice with half of the butter.
Heat a dry small nonstick skillet over medium heat.
Place 1 slice of bread, butter side down, in the pan and top with the cheeses, alternating types. Mozzarella and then white cheddar, then mozzarella, then the final piece of white cheddar.
Arrange the second slice of bread, butter side up, on top.
Cook the sandwich until golden brown underneath, about 2 minutes. Carefully flip the sandwich using a flat spatula. Gently press down on the sandwich with that spatula to encourage even browning and help the cheese melt.
Cook until the second side is golden brown and the cheese is melted, about 2 minutes.
Once golden brown on both sides and the cheese has melted, remove from the heat. Cut the sandwiches in half and serve immediately.