To roast the tomatoes: Preheat the oven to 350 degrees F. Season the tomatoes with oregano, salt and pepper, to taste. Drizzle with oil and arrange them on a cooling rack over a baking sheet. Roast until then are golden, 20 to 25 minutes.
Raise the oven temperature to 400 degrees F when the tomatoes are roasted. Unfold 1 puff pastry sheet onto a clean cutting board. Prick the dough evenly, multiple times, with a fork. Using a 2-inch pastry cutter, cut out 12 circles and arrange them on a parchment lined baking sheet. Repeat with the second sheet of pastry. Bake until they are nicely puffed and golden, about 8 minutes.
To assemble: Cut the puffs in half. Spoon a generous teaspoon of marinara sauce into the bottom of half of the puffs and sprinkle with Parmesan. Put a roasted tomato on top of each followed by the basil and a sprinkling of the mozzarella. Top with the other puff pastry halves and arrange them on a baking sheet. Bake for another 5 minutes, giving the cheese time to melt. Transfer the bites to a serving platter and serve warm.