Preheat the oven to 400 degrees . On a work surface, unfold the puff pastry and cut into 6 rectangles. Place on a parchment-paper-lined baking sheet and sprinkle with sugar. Bake until golden-brown, about 15 minutes, then lower the heat to 325 degrees and bake for 5 minutes more. Let cool.
Meanwhile, toss together the mangoes and 2 tablespoons sugar; let stand for 5 minutes.
Mix the yogurt with the remaining 2 tablespoons sugar and the vanilla. Using a serrated knife, halve the baked pastries horizontally. Spread the yogurt-vanilla mixture on the bottom halves, spoon the mango on top and cover with the remaining pastry halves.