Whisk the flour, cocoa powder, baking soda, baking powder and salt in a medium bowl. Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 5 minutes. Add the egg and vanilla; beat until creamy, about 2 more minutes. Reduce the mixer speed to low; gradually add the flour mixture and beat until just incorporated, about 2 more minutes.
Divide the dough between 2 sheets of plastic wrap; form into disks. Wrap and refrigerate until firm, at least 1 hour.
Roll out the dough between floured sheets of parchment paper until about 1/4 inch thick. Chill 2 more hours.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Mist 2 baking sheets with cooking spray. Cut the dough into shapes; gather the scraps and reroll once to cut out more cookies. Arrange 2 inches apart on the prepared baking sheets. Bake, switching the pans halfway through, until set, about 12 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Make the icing: Beat the confectioners' sugar, meringue powder and 3 to 4 tablespoons water in a bowl with a mixer on high speed until smooth and thick. Transfer to a resealable bag and snip a small corner. Pipe onto the cookies; sprinkle with coarse sugar. Let set 30 minutes.