Make the cookies: Whisk the flour, cocoa powder, salt and baking powder in a medium bowl. Beat the butter and sugar in a large bowl with a mixer on medium-high speed until light and fluffy, 3 to 4 minutes. Beat in the egg yolk and vanilla and peppermint extracts, about 30 seconds. Reduce the mixer speed to low; beat in the flour mixture until just combined. Pat the dough into a rectangle and place between 2 sheets of parchment paper. Roll out into an 8-by-13-inch rectangle, about 1/4 inch thick. Refrigerate until firm, at least 1 hour.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Cut the dough into 1-by-2 1/2-inch rectangles. (If the dough is too soft, refrigerate 15 minutes.) Arrange 1 inch apart on the prepared pans.
Bake, switching the pans halfway through, until the cookies are set around the edges, 18 minutes. Let cool 10 minutes on the pans, then transfer to racks to cool completely.
Make the icing: Melt the chocolate with the heavy cream and butter in a medium heatproof bowl set over a small saucepan of simmering water, stirring (do not let the bowl touch the water). Remove the bowl from the pan and let cool 5 minutes. Spread on the cookies. Sprinkle with peppermints.