Thaw the puff pastry according to the package directions (overnight in the fridge is best). Keep it chilled until ready to work with so it puffs well in the oven.
Preheat the oven to 400°F (204°C). Line a baking sheet with parchment paper if desired.
Place the flattened puff pastry sheet on a cutting board and cut it in half to make two squares.
Using a paring knife, lightly score a 1/2-inch border around each square by cutting about halfway to three-quarters through the dough, taking care not to cut all the way through.
In a medium bowl, stir together the softened butter, minced garlic, parsley, salt, and black pepper until combined.
Add the shredded mozzarella and grated Parmesan to the butter mixture and stir until evenly incorporated.
Divide the cheese-garlic mixture evenly between the two pastry rectangles, spreading it inside the scored borders and leaving the edges clear.
Transfer the filled pastry squares to the prepared baking sheet and bake for about 15 to 18 minutes, or until the edges are puffed and golden and the cheese is lightly browned. Check early to avoid overbrowning.
Remove from the oven and serve immediately.