In the bowl of a stand mixer or large mixing bowl, cream the butter and sweetener together until soft and fluffy.
Add the egg and vanilla. Mix well.
Add the almond flour, coconut flour, baking powder, and salt. Mix until well combined. Scrape the sides and bottom of the bowl with a rubber spatula and mix again if needed.
Transfer the dough to a large piece of plastic wrap. Use your hands to roll the dough into a log about 1 1/2 to 2 inches in diameter. Wrap the log tightly in the plastic wrap.
Chill the dough log in the refrigerator for a minimum of 30 minutes. If desired, you can also flash-freeze the dough for an additional 30 minutes to make it easier to handle.
When you're ready to bake, preheat oven to 350°F. Line a cookie sheet with parchment paper.
Remove the dough log from the plastic wrap and cut it into 24 even slices.
Lay the slices about 1 inch apart on the baking sheet. Bake in batches to avoid overcrowding the baking sheet. Keep the unbaked dough in the fridge between batches.
Bake until the edges of the cookies are golden brown, their surfaces go from glossy to matte, and the centers are set but still soft, about 10 to 13 minutes.
Remove the cookies from the oven and let them cool on the baking sheet for 2 to 3 minutes. Then transfer them to a wire rack.
Let the cookies cool completely on the wire rack before eating or frosting them.