Preheat the oven to 350°F.
In a large bowl, beat the room temperature butter and sugar together with an electric mixer until they mixture is creamy and light. You can't beat it too much.
Add the egg, vanilla extract, and peppermint extract and beat until combined.
In a separate large bowl, sift together the flour, cocoa powder, baking powder, and salt. Whisk together to fully combine.
Add the flour and cocoa powder mixture to the wet ingredients, a little at a time, beating with the mixer until well blended.
Cover a large flat surface with a sheet of parchment paper. Divide the cookie dough in half and place half on the parchment paper.
Place another piece of parchment paper on top of the dough and roll it out until it's ¼-inch thin. Repeat with the other half of the dough.
Keeping the rolled dough between the sheets of parchment, refrigerate for at least two hours.
After the dough has chilled, remove one half at a time and use a 2-inch round cookie cutter to cut 28 circles from each half, re-rolling the scraps as necessary to use all the dough. As you cut the cookies, transfer them to a cookie sheet lined with parchment paper. Because the cookies are so small, and they don’t expand much in the oven, you can place them close together on the cookie sheet. Give each cookie just a little bit of room to expand.
Bake each sheet of cookies at 350ºF for eight minutes or until fully cooked.
Remove cookies from the oven, allow to cool on the sheet for 5 minutes, then transfer to a cooling rack and fully cool to room temperature.