Line a baking sheet with parchment paper.

In a bowl toss together the apples, lemon juice, brown sugar, granulated sugar, salt, cinnamon, nutmeg, and butter until well combined. Place the apple mixture in medium saucepan and cook over medium heat, stirring occasionally. Continue cooking until apples are soft and mixture has reduced to a jam-like consistency, about 6 to 8 minutes. Remove from heat and let cool to room temperature.

When the apple mixture has cooled, whisk together the egg and milk in a small bowl to make an egg wash.

Spread the sheet of puff pastry on a lightly floured surface. Dust the pastry lightly with flour and, using a rolling pin, roll the pastry into a 12-inch square. Using a sharp paring knife, trim the edges and cut the dough into four equal squares. Using a pastry brush, brush the edges of each square lightly with the egg wash.

Spoon one quarter of the apple mixture in the middle of each square, and fold the dough in half to form a triangle. Seal the edges by pressing them together with the tines of a fork. Transfer to the baking sheet. Poke a few holes in the top of each turnover with the tip of a paring knife.

Cover the baking sheet with plastic wrap and refrigerate for 15 to 30 minutes to firm slightly. Meanwhile, preheat the oven to 400F.

Remove the turnovers from the refrigerator, and remove the plastic wrap. Brush the top of each turnover with the egg wash.

Bake until golden brown and flaky, about 20 minutes. Transfer to a wire rack to cool slightly. Serve warm, with a scoop of ice cream, if desired.