Make the cookies: Preheat the oven to 325˚. Whisk the flour, cocoa powder, baking powder and salt in a medium bowl. Beat the butter and sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 5 minutes. Beat in the vanilla and egg until just combined. Reduce the mixer speed to medium low and beat in the flour mixture until just combined.
Fit a cookie press with a wreath disk; fill with the dough according to the manufacturer’s instructions. Press cookies 1 inch apart onto 2 unlined baking sheets.
Bake the cookies, 1 pan at a time, until crisp and set around the edges, 18 to 22 minutes. Immediately remove the cookies to a rack using an offset spatula and let cool completely. Repeat with the remaining cookies.
Once the cookies have cooled, make the icing: Combine the chocolate, heavy cream and butter in a medium heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water). Stir until melted and smooth, 2 to 4 minutes. Remove the bowl from the pan and stir in the peppermint extract; let cool 5 minutes.
Transfer the icing to a resealable plastic bag, snip off a corner and drizzle over the cookies. Immediately sprinkle with crushed candy canes. Let set, about 30 minutes.