Step 1Heat the oven to 450 degrees. Set aside a large greased baking sheet.
Step 2On a floured surface, roll the puff pastry in a 13-inch square, as thin as possible, about 1/16-inch thick. Use a 6-inch plate to outline 4 circles on the pastry, then cut them out. Transfer them to the baking sheet in a single layer.
Step 3Arrange a tight circle of slices from each apple half on each pastry round. Combine the sugar and cinnamon in a small dish. Sprinkle 1 tablespoon of the mixture evenly over each tart. Dot each evenly with 3/4 teaspoon of butter. (This can be prepared up to 6 hours in advance and refrigerated or frozen up to 1 week, wrapped airtight.)
Step 4Bake the tarts until browned around the edges, about 15 to 18 minutes. Use a metal spatula to transfer them to dessert plates. Serve immediately with small scoops of ice cream placed in the center.