Heat oven to 400°. Line two baking sheets with parchment. On a lightly floured work surface, unfold pastry and cut into 4 rectangles. Cut each rectangle in half to make 8 squares. Use your hands to gently pull, press, and shape each pastry into a rough circle. (Keep remaining pastry chilled while working with one square at a time.) Transfer to parchment and prick all over with a fork.
Spread 1 tablespoon mascarpone evenly over each pastry, leaving a 1-inch border. Arrange peach slices on top and sprinkle with a heaping teaspoon sugar. Wet fingers to fold border up and crimp along the edges. Dot with butter. Bake until pastry is golden brown, rotating pan halfway through baking time, about 30 minutes. Let cool before serving.