Preheat the oven to 425 degrees F (220 degrees C). Spread each sheet of puff pastry onto a piece of lightly floured parchment paper. Use a rolling pin to roll pastry into even rectangles. Lift pastry (with parchment paper) onto baking sheets.
Pierce puff pastry all over with a fork. Beat egg with water in a small dish; brush pastry all over with egg wash.
Bake in the preheated oven for 10 minutes.
Meanwhile, pulse tomatoes, ½ teaspoon of the olive oil, garlic, and a pinch of salt together in a food processor until pureed.
Carefully press pastry crusts down to deflate them. Spread half the tomato sauce evenly to within about an inch of edges of each crust; drizzle with remaining olive oil. Top evenly with mozzarella cheese and Parmigiano-Reggiano cheese.
Bake until cheese is melted and crust edges are golden, about 5 minutes more. Remove from the oven and scatter basil leaves over the top. Slice and serve hot.
