Grind the freeze-dried apples (see note) to a fine powder in the bowl of a food processor. Remove any larger pieces of apple that remain.
In a medium bowl, combine the apple powder, flour, cinnamon, and baking powder. Set aside.
In the bowl of a stand mixer, combine butter, sugar, and brown sugar. Beat until light and fluffy. Scrape down the sides of the bowl.
Add egg and mix to combine, scraping down the bowl again as needed.
Add the apple/flour mixture gradually, mixing just until incorporated.
Turn the dough out onto your work surface and divide in half. Shape each half into a disc about 1 inch thick, wrap each in plastic, and chill for 30 minutes.
After chilling, working with half the dough at a time, roll the dough out to about ¼ inch thickness between two sheets of waxed paper. Transfer the rolled-out dough to a cookie sheet and refrigerate another 30 minutes.
Line two cookie sheets with parchment or silpats (see note). Set aside.
After chilling again, cut the dough out into desired shapes (I chose a 2.5-inch fluted round cutter). Place the cut-outs on the prepared cookie sheets, and return the refrigerator while the oven preheats to 375° F.
After the final chilling, bake cookies for 7-9 minutes, or until just barely golden at the edges. Allow the cookies to cool on the pan for at least 10 minutes before removing to a wire rack to cool completely.
Once cookies are completely cool, sandwich with dulce de leche.
Once baked, the cookies will be good in an airtight container for about three days. For longer storage, store the baked (and cooled!) cookies in an airtight container in the freezer for up to six months, then just thaw and fill!Enjoy!
Notes
Note on freeze-dried apples: Make sure that you're using *freeze-dried* apples and not just dried. Freeze-dried apples will be crunchy and allow you to grind them into a powder. After pulsing the apples into a powder and removing any large pieces that remained, I had about ¾ cup of apple powder.Note on silpats: I like lining my pans with silpats because they can be used over and over again and because the cookies brown more slowly. If you line your pans with parchment paper, just keep in mind that they may brown up a little quicker than mine did.