Preheat oven to 425°F; line a large baking sheet with parchment. On a lightly floured countertop, roll out 1 sheet of thawed storebought frozen puff pastry to a 14-by-12-inch rectangle. Transfer pastry to baking sheet. Brush edges lightly with water; fold them in to form a 1/2-inch border, then press with a fork to seal all around. Prick pastry all over with fork.
Mix 3 Tbsp. chopped fresh basil with 3/4 cup grated fresh mozzarella and 1/4 cup grated Parmesan. Spread 2 Tbsp. jarred blackolive tapenade over inside of pastry. Sprinkle with cheese-and-basil mixture, then arrange 3 thinly sliced tomatoes decoratively on top. Sprinkle with salt and pepper. Bake until crust is deep golden and cheeses are melted, 20 to 25 minutes. Sprinkle with 1 Tbsp. fresh basil and serve.