Combine the flour, cornstarch, cocoa powder, salt, baking powder, and baking soda.
In a separate bowl, beat together the sugar and butterĀ until the mixture is fluffy. Add milk, egg, and vanilla and incorporate. Slowly add the flour mixture, mixing until just combined.
Divide the dough into two discs and wrap in plastic wrap. Chill in the refrigerator for 30-45 minutes until the dough is cold and firm.
Preheat the oven to 350F. Line baking sheets with parchment paper.
Pull one disc of the dough from the fridge and on a lightly floured work surface, roll out to 1/4" thick. Use a small round or floral cutter to cut the cookies and place them onto a cookie sheet, leaving a little bit of space between each. Repeat with remaining dough.
Bake the cookies for 15 minutes, then let the cookies cool in the pan for about 5 minutes before transferring to a wire rack to cool completely.
While cookies are cooling, spread out a large piece of waxed paper onto the countertop or onto cookie sheets.
Combine shortening, chocolate, and peppermint extract to a glass bowl. Set a pot on the stove with the water boiling and rest the glass bowl over top. Melt the chocolate, stirring frequently until the mixture is smooth. Turn the heat down to very low, just to maintain smoothness.
Drop one cookie at a time into the melted chocolate. Use a fork to flip over the cookie and lift it out of the chocolate, shaking to let any excess chocolate drop from the cookie. Transfer each cookie to wax paper and allow to fully dry and harden.