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.