Prepare the Puff Pastry: Preheat your oven to 400°F (200°C). On a lightly floured surface, roll out the thawed puff pastry sheet to fit your baking sheet. Once rolled out, carefully place the pastry onto a parchment-lined baking sheet. Using a sharp knife, score a 1-inch border around the edge of the pastry, ensuring not to cut all the way through. This will create a raised edge as it bakes. Brush the border with your beaten egg to achieve a glossy finish.
Prepare the Topping: In a mixing bowl, combine the olive oil, minced garlic, dried oregano, salt, and pepper. Mix well to combine. Layer the thinly sliced tomatoes within the scored border, ensuring they are evenly distributed. Next, sprinkle the mozzarella and Parmesan cheeses over the tomatoes. Finally, drizzle the olive oil mixture over the layered ingredients, making sure everything is nicely coated.
Bake: Place the baking sheet in the preheated oven and bake for 20-25 minutes or until the pastry is golden brown and the cheese is bubbly. Keep an eye on it to avoid over-browning. Once baked, remove it from the oven and let it cool for about 5 minutes.
Serve: Just before serving, scatter the chopped fresh basil over the top as a garnish. For an extra touch, drizzle with balsamic glaze if desired. Slice the pastry into squares or wedges and enjoy it while warm or at room temperature!