Preheat the oven to 375F. Cut the puff pastry and fit it into 6 lightly greased individual tartlet molds. Trim the excess pastry from the edges, leaving 1/2 inch overhang to fold over the filling.
In a medium bowl, stir the granulated sugar, brown sugar, cornstarch, and ground cinnamon until they are thoroughly blended. Gently toss the sliced apples in the sugar mixture.
Divide the sugar-spiced apples among the prepared tartlet molds and fold the edges over the apples. Sprinkle the pastries with the coarse sugar. Arrange the tartlets on a baking sheet and bake them in the preheated oven for 35 to 40 minutes, until they turn golden brown and the apples are tender.
Allow the tartlets to cool for at least 30 minutes and serve them warm or at room temperature.