Special equipment: a 2 1/2- to 3-inch round cookie cutter
Position oven racks in the upper and lower thirds of the oven; preheat the oven to 350 degrees F. Line two rimmed baking sheets with parchment paper.
Whisk the flour, cocoa powder, baking soda and salt together in a medium bowl.
Beat the butter and sugar together in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 3 minutes. Add the egg, vanilla and peppermint extract if using, and beat until just combined. Add the flour mixture and beat on low speed until combined. Turn the dough out onto a work surface and knead several times with your hands to bring the dough together. Divide the dough in half and wrap each half in plastic wrap. Refrigerate to chill slightly, about 10 minutes (or refrigerate half the dough and freeze the other half for later use).
Generously dust a work surface with flour. Place one half of the dough on top and dust with more flour. Roll out into a round slightly less than 1/4-inch thick and about 14 inches in diameter, adding a little more flour as necessary to prevent sticking. If the dough cracks, press it back together with your fingers and smooth with the rolling pin. Use a 2 1/2- to 3-inch round cookie cutter to cut out cookies. Transfer the cookies one of the prepared baking sheets, leaving about 1/2 inch between each cookie. Gather the scraps and refrigerate. Repeat with the second half of the dough.
Bake, rotating the position of the baking sheets halfway through, until slightly puffed and a shade darker on the underside, about 10 minutes. Let cool on the baking sheets for 5 minutes, then transfer to a rack to cool completely (the cookies will crisp as they cool).
Gather all of the dough scraps together and roll out in the same manner. Cut out cookies and bake. Discard any remaining scraps.