In a medium saucepan, stir together the sugar, brown sugar, salt, cinnamon, nutmeg, allspice, and 1 tablespoon plus 1 teaspoon cornstarch. Add the apples.
Bring the apples to a simmer over medium-low heat, stirring frequently, until the apples soften and the liquid thickens, about 15 minutes. Remove from heat and let cool.
Thaw puff pastry according to the package directions.
Preheat the oven to 400°F, and line a baking sheet with parchment paper.
On a lightly floured surface, roll the pastry to a 9-by-9-inch square. Cut into 9 squares and roll each to about 5 inches.
Place 2 tablespoons of the apple mixture just off center of each square.
Dampen the outer pastry edge with water and fold the dough over the filling to make a rectangle. Pinch edges firmly and crimp them with a fork.
Transfer to prepared baking sheet. Gently pierce the tops with a fork to allow steam to escape, and lightly spray with cooking spray (or brush with egg wash).
Bake for 15 minutes, until puffy and golden. Let cool slightly and dust with confectioners’ sugar (if using) before serving.