Preheat oven to 425°F. In a medium bowl, combine plums, sugar and cinnamon. Let stand, tossing occasionally, until sugar dissolves.
Roll out one sheet of puff pastry on a lightly floured surface to an 1/8-inch thickness. Use a 6-inch plate as a guide to cut out 2 rounds; discard excess dough. Repeat with remaining puff pastry sheet.
Place pastry rounds at least 2 inches apart on a large baking sheet. Arrange plum slices on pastry sheets in concentric circles, overlapping them slightly. Bake until pastry is golden brown and fruit is soft, 20 to 25 minutes. Let tarts cool slightly. Serve warm.