In a bowl, combine the diced apples, brown sugar, cinnamon, nutmeg, lemon juice, and cornstarch. Stir to coat and let sit a few minutes to thicken.
On a lightly floured surface, unfold the puff pastry, roll slightly to even it, and cut into 4 equal squares.
Spoon the apple mixture onto one half of each square, leaving a small border. Fold into a triangle, press the edges, and crimp with a fork to seal.
Brush the tops with beaten egg and sprinkle with sugar.
Preheat the air fryer to 350°F (175°C). Place turnovers in a single layer, not touching, and air fry 10–12 minutes until golden and puffed. Cook in batches if needed.
Cool 5 minutes on a wire rack. Whisk powdered sugar with milk and drizzle over the top, if using. Serve warm.