In a saucepan over medium heat, combine the apples and the butter. Coat the apples with the butter and cook until softened, about 5 minutes.
Add the brown sugar, honey, cinnamon, nutmeg, and lemon juice to the pan. Stir and simmer for 3 minutes, until slightly caramelized. Remove from heat and allow to cool completely.
Spread out the puff pastry sheet and cut it into 4 squares.
Beat together the egg and water in a bowl.
Scoop 1/4 of the apple mixture onto each puff pastry square. Brush the egg wash around the edge of each square. Fold it over and press the edges to seal. Press the edges with a fork.
Brush the tops of each turnover with the egg wash and use a sharp knife to cut three slits in the top.
Turn the air fryer on to 350° F(175°C) and preheat for 3 minutes. Place as many turnovers as you can fit without overlapping. Cook for 12-15 minutes in the preheated air fryer, just until the tops are lightly browned. Transfer to a wire rack to cool slightly.
For the optional glaze, mix together the powdered sugar, cream and vanilla until smooth. Add additional cream to reach your desired consistency. Pour over the turnover before eating.
Notes
Use good baking apples to make these. Some good ones to try are Honeycrisp, Granny Smith, Cortland, McIntosh, and Jonagold. I always say, use an apple that you would enjoy eating. Avoid soggy, tasteless apples.
If you're allergic to egg, you can spray the tops of the pastries with a light spritz of oil spray instead of the egg wash.
Be sure to thaw the puff pastry in the refrigerator over night before you use it. It shouldn't be frozen when you use it or it won't fold or cook properly.
All air fryers have different wattage and may cook at different rates. You may need to adjust the time or temp for the best results.