Mix the softened butter, finely grated Parmesan, garlic powder, and black pepper in a small bowl until evenly combined.
Spread a thin, even layer of Parmesan butter on one side of every sourdough slice, taking it all the way to the edges.
Place two slices buttered-side down. Layer each with one slice cheddar, two slices mozzarella, and one more slice cheddar. Close with the remaining bread, buttered-side up.
Preheat a large skillet over medium-low for 2 minutes. Add the sandwiches and cook for 3 to 4 minutes until golden. Cover for the final minute to help melt the cheese.
Carefully flip and cook for another 3 to 4 minutes, covering again if needed, until both sides are crisp and the cheese is fully melted.
Transfer to a cutting board, rest for 1 minute, slice diagonally, and serve immediately.