Preheat oven to 425 degrees F. Melt butter in large non-stick skillet over medium-heat. Add apples; sauté until apples begin to brown, about 5 minutes. Add sugar, cinnamon and nutmeg and sauté until golden brown, about three minutes longer. Set aside to cool.
On a lightly floured board, one at a time, roll the pastry sheets out to approximately 12 inches square. Trim the outer edges (this is essential for the puff pastry) and cut each sheet into four 6-inch squares, making eight in all. Place a generous 1/4 cup of fruit filling just off center in each square. Moisten the edges of two adjacent sides with water, fold the opposite sides over to make a triangle and press with fingers to seal. Crimp the edges closed with the tines of a fork. Repeat with the remaining turnovers and chill thoroughly. Beat together the egg white and water and brush lightly on top of the turnover without letting the egg wash drip down the cut sides. Sprinkle the tops with sugar. Place the turnovers on cookie sheets lined with parchment paper or foil.
Bake until they are puffed and golden, about 20 minutes. Cool for 10 minutes and serve.