You can use either natural or Dutch-processed cocoa powder here. The natural cocoa will give your cookies a lighter color and flavor, while the Dutch-processed cocoa will give the cookie a more intensely chocolaty flavor and deep brown color. Similarly, if you want a cookie more in the style of the inspiration, use sweeter semisweet chocolate. But if you prefer less sweet chocolate, use bittersweet, preferably no more than 70%.
Heat the oven to 325 degrees. Line three baking sheets with parchment paper.
In a large bowl, combine the butter and sugar and use a wooden spoon to cream the two together until paler in color and fluffy, about 1 minute. Stir in the vanilla and ½ teaspoon peppermint extract, then add the egg and stir until smooth. Place a sieve over the bowl and add the flour, cocoa powder, salt and baking soda to the sieve. Sift the dry ingredients into the bowl, then stir slowly until they’re absorbed into a stiff dough.
Dust a clean work surface liberally with flour, then scrape the dough onto the flour. Sprinkle with a little more flour, then flatten the dough, either with your hands or a rolling pin, until ⅛-inch thick. Using a 2-inch round cutter, stamp out rounds of dough and transfer them to the prepared baking sheets, leaving about ½ inch between each (these don’t spread at all while baking). Reroll scraps as many times as necessary until you no longer have any dough left.
Bake the cookies, one sheet at a time, until dry to the touch and set in the center, 13 to 15 minutes. Let the cookies cool completely on their baking sheets.
To glaze the cookies, melt the chocolate in a heatproof bowl. If using a microwave, heat in 1-minute bursts at 50% power, stirring between each burst, until the chocolate is just melted. If using a double boiler, set the bowl over a small pan of simmering water (make sure the bottom of the bowl does not touch the water) and stir until the chocolate is fully melted. Add the shortening and the remaining ½ teaspoon peppermint extract to the chocolate and stir until melted and smooth.
Add one cookie to the bowl of melted chocolate and use a spoon to pour chocolate over the top. Using the tines of a fork, lift the cookie out of the chocolate and lightly rap the neck of the fork on the side of the bowl repeatedly until the excess chocolate drips off the cookie and back into the bowl. Transfer the cookie back to its baking sheet, letting it fall gently off the tip of the fork’s tines. Repeat with the remaining cookies and chocolate, re-melting the chocolate as needed if it begins to thicken or cool too much.
Let the cookies stand until the chocolate is firm to the touch before removing from the baking sheets to serve.
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