Generously flour a working surface. Roll the puff pastry into a 9" x 15" rectangle. Cut the pastry into 6 equal squares. Refrigerate until needed. Bring the water and sugar to a boil in a large saucepan. Peel and core the apples; cut into 1⁄2" chunks. Toss the apples with the cinnamon, ginger, nutmeg, and lemon juice, then add to the sugar mixture and simmer until the liquid thickens, 5 to 7 minutes. Set aside to cool. Preheat the oven to 425°F. When the apple mixture cools, spoon 2 to 3 tablespoons of apples onto the puff pastry squares, off center. Fold the pastry over the apple filling to make a triangle and seal the perimeter by pinching the edges or pressing the tines of a fork into the dough. Place the turnovers on a baking sheet and brush with the egg white. Bake until the dough is crispy and golden brown, 12 to 15 minutes.