Make the cupcakes: Position a rack in the middle of the oven and preheat to 350 degrees F. Line two 12-cup muffin pans with paper liners. Put the butter, cocoa powder and 3/4 cup water in a medium microwave-safe bowl, cover with plastic wrap and microwave until the butter melts, about 2 minutes. Whisk to combine, then whisk in the brown sugar.
Whisk the flour, baking powder, baking soda and salt in a large bowl. Whisk in the warm cocoa mixture. In another bowl, whisk the buttermilk, eggs and vanilla; stir into the batter until just combined but don't overmix.
Divide the batter among the prepared cups, filling each about three-quarters of the way. Bake until the cupcakes spring back when touched, 30 to 35 minutes. Let cool in the pans 10 minutes, then transfer to racks to cool completely.
Make the topping: Beat the confectioners' sugar, corn syrup, shortening and peppermint extract with a mixer until a tight paste forms. Gather into a ball, place between two sheets of parchment paper and roll out to 1/4 inch thick (microwave 15 seconds to soften, if necessary). Use a 21/2-inch round cutter or juice glass to cut into disks; reroll the scraps. Place a peppermint disk on top of each cupcake.
Make the glaze: Put the chocolate, corn syrup, butter and 1/4 cup water in a microwave-safe bowl, cover with plastic wrap and microwave on 50 percent power until the chocolate melts, about 2 minutes. Whisk to combine, then cool slightly. Spread the glaze over the peppermint topping, leaving some peppermint exposed. Refrigerate until set, at least 20 minutes. Serve cold.