Put the butter and granulated sugar in a stand mixer fitted with a paddle attachment. Beat until light and fluffy, about 2 minutes. Add the egg and peppermint extract. Beat until just combined.
Whisk together the flour, cocoa powder, baking powder and salt in a medium bowl. Pour half of the dry ingredients into the butter mixture and beat on low speed until combined. Repeat with the remaining dry ingredients until a dry dough comes together. Collect the dough and press into 2 mounds. Place each mound a third of the way down on a big piece of parchment paper. Fold the top third of parchment paper over the mound. Then use a ruler to push back on the mound while pulling the bottom of the parchment paper to shape it into a 1 1/2-inch log. Wrap both logs in plastic wrap and chill in the refrigerator for at least 2 hours or up to overnight.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Remove the logs from the refrigerator and put onto a cutting board. Slice the logs into 1/4-inch-thick rounds. Place on the prepared baking sheets.
Bake the cookies until the tops look dry, 8 to 9 minutes, rotating halfway through. Let the cookies to cool on the baking sheets for 3 minutes before moving to a wire rack to cool completely.
Put the confectioners' sugar into a fine mesh strainer and sprinkle gently over the tops of the cookies before serving.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)