Preheat the oven to 375° and line a large baking sheet with parchment paper. On a lightly floured work surface, roll out the puff pastry 1/8 inch thick. Cut the pastry into six 6-inch squares and transfer to the baking sheet; freeze for 5 minutes.
In a bowl, toss the plums with the 1/4 cup of sugar and the cornstarch and cardamom. Spoon the fruit and juices onto the center of each puff pastry square. Working with 1 square at a time, bring up all 4 corners and pinch them together in the center. Pinch the seams closed at the bottom corners to trap any juices, leaving the tops open to vent steam. Brush the pastries with the egg and sprinkle with sugar. Bake in the center of the oven for about 30 minutes, until puffed and golden. Let cool slightly and serve with scoops of vanilla ice cream.