In a large saucepan, melt the butter over medium heat. Then add the apples, cinnamon, brown sugar, lemon juice. Stir to combine.
In a small bowl, whisk together the cornstarch with 1 tablespoon of water to form a slurry. Pour the slurry into the pan with the apple mixture and continue to stir and cook until the mixture has thickened up a bit.
Meanwhile, unroll the puff pastry sheets and use a 3-inch cookie cutter or pastry cutter to cut out circles.
When the apples have cooked a bit and become slightly tender and the mixture has thickened, turn off the heat and remove the pan.
Place one pastry round on a paper towel, cutting board, or parchment paper. Use a fork to poke two sets of holes lightly near the center. This helps prevent the pastry from "puffing up" too much on the bottom as it air fries.
Place one tablespoonful of filling into the center of a pastry round, leaving some space (a little less than about ½-inch) around the edges. Using your fingers, wet the edges with water. Then place another pastry round on top and press down to seal the edges all the way around.
Using the tines of a fork, gently press down along the edges of the pastry to help further seal the two pastry rounds together. This also creates a cute pattern around the edges! Then, using a small sharp knife, cut two slits in the middle of the top pastry. Repeat until you use all of the pastry rounds. You may have some filling leftover - save this for topping your Apple Cinnamon Pancakes for breakfast!
In a small bowl, beat together the egg and 1 tablespoon of water to create an egg wash. Using a pastry brush, brush enough egg wash just to cover the tops of each filled pastry. Place the filled pastries onto the prepared air fryer basket or trays.
Air fry at 350°F for 7 to 9 minutes, just until the puff pastry has fully cooked and the tops are golden brown.
Make the caramel sauce
While the pastries are air frying, make the caramel sauce. In a heavy 3-quart saucepan, melt the sugar and water over medium-high heat. Whisk constantly until the sugar is dissolved. Once the sugar mixture reaches a low boil, stop whisking and allow the mixture to continue boiling until it reaches a medium-dark amber color. This step can take a while, but be sure to watch the mixture carefully: if you step away, this is where it could burn. Every once in a while, swirl the pan to ensure even boiling.
As soon as the sugar mixture reaches that dark amber color, take it off the heat and immediately add the butter. Whisk briskly until the butter melts completely. Slowly and carefully pour in the heavy cream and whisk to combine.
The mixture may bubble as you whisk, which is normal.Keep whisking the mixture until it is a gorgeous, luscious caramel sauce.
Drizzle or spoon caramel sauce over each cooked apple hand pie. Pour the remaining caramel sauce into a glass jar and cool, uncovered, to room temperature. Keep this jar in the fridge to use for other desserts!
Notes
I use my Ninja Foodi Deluxe XL to air fry these apple hand pies, using the Air Crisp function. Place the deluxe reversible rack into the Ninja Foodi pot in the lower position, place half of the apple hand pies on the rack, then place the additional top rack in and place the remaining hand pies on the top rack. If using a tray-style air fryer or basket air fryer, you may want to place parchment paper over the bottom of the trays or basket. Storage: these keep in an airtight container at room temperature overnight, or in the refrigerator for up to 3 days. To reheat: you can reheat refrigerated apple hand pies by air frying them at 300°F for 5 to 7 minutes.