Slice the apples: Peel and core the apples. Cut them into ¼- to ½-inch rounds and widen each center hole to about 1 inch. Brush both sides with lemon juice and pat dry.
Cut the puff pastry: Unfold the cold pastry on a lightly floured surface, press the seams closed and cut into ½-inch strips.
Wrap the apple rings: Loop 2 to 3 pastry strips through the center and around each apple slice, overlapping slightly. Tuck the ends underneath.
Add the egg wash: Whisk the egg white and water. Brush over both sides of each wrapped ring.
Coat in cinnamon sugar: Mix the sugar and cinnamon in a shallow bowl. Press each ring into the mixture to coat both sides.
Prepare the air fryer: Line the basket with parchment or spray it lightly. Arrange the rings in a single layer, not touching.
Air fry: Cook at 380°F for 6 to 8 minutes. Flip and cook 2 to 4 minutes more, until the pastry is deep golden and flaky.
Finish and serve: Brush with melted butter if using. Cool 5 minutes, then serve warm with caramel sauce or ice cream.