Cut puff pastry sheets into 6 (5-inch) circles, and place on baking sheets. Spread evenly with butter, and sprinkle with 2 tablespoons sugar. Lightly grease the bottom of a baking sheet, and place directly on pastry circles.
Bake at 400° for 10 minutes or until golden brown. Remove to wire racks to cool.
Sprinkle 1 cup sugar in a large skillet; cook over medium heat, stirring constantly, until golden brown. Stir in apple slices and cinnamon; cook, stirring occasionally, until tender. Remove apples with a slotted spoon.
Stir brandy into sugar mixture; cook, stirring occasionally, 10 to 15 minutes or until thickened.
Place pastry circles on individual dessert plates. Arrange apple slices on top, and drizzle with brandy mixture. Top tarts with vanilla ice cream, and serve immediately.