Melt butter in a large skillet over medium heat. Drain water from apples, and place them into the hot skillet. Cook and stir for about 2 minutes.

Add brown sugar, and cinnamon, and cook, stirring, for 2 more minutes.

Stir together cornstarch and 1 tablespoon water. Pour into the skillet and mix well. Cook for another minute, or until sauce has thickened. Remove from heat to cool slightly.

Preheat the oven to 400 degrees F (200 degrees C).

Unfold puff pastry sheets, and repair any cracks by pressing them back together. Trim each sheet into a square.

Then cut each larger square into 4 smaller squares. Spoon apples onto the center of each square. Fold over from corner to corner into a triangle shape, and press edges together to seal.

Place turnovers on a baking sheet, leaving about 1 inch between them.

Bake for 25 minutes in the preheated oven, until turnovers are puffed and lightly browned. Cool completely before glazing.

To make the glaze: Mix confectioners' sugar, milk, and vanilla together in a small bowl. Adjust the thickness by adding more sugar or milk if necessary.

Drizzle glaze over the cooled turnovers.