Melt the butter in a skillet over medium heat. Add the apples, brown sugar, cinnamon, and lemon juice, then cook for 5 to 7 minutes until slightly softened.
Stir the cornstarch and water together in a small cup. Pour the mixture into the skillet and cook for another minute, until the juices thicken, then cool completely.
Preheat the oven to 400°F or 200°C and line a baking sheet with parchment. Unfold the pastry sheets and cut each one into four equal squares.
Spoon a small amount of cooled filling onto one half of each square. Fold the pastry diagonally over the apples and press the edges together.
Seal the edges with the tines of a fork and cut a small slit in the top of every turnover. Whisk the egg with milk and brush the mixture over the pastry.
Sprinkle with coarse sugar when desired and bake for 18 to 22 minutes. The turnovers are ready when deeply golden, puffed, and crisp around the edges.
