Whisk the flour, 1/4 cup cocoa powder, the baking powder and salt in a bowl. Beat 1 stick butter and the granulated sugar in a large bowl with a mixer on medium speed until light and fluffy, 3 to 5 minutes. Beat in the egg and 1/2 teaspoon vanilla. Reduce the mixer speed to medium low; beat in the flour mixture until just incorporated. Divide the dough between 2 sheets of plastic wrap; form into flat rectangles. Wrap and 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. Working with 1 piece at a time, roll out the dough into a 10-inch square (1/8 inch thick) on a lightly floured surface. Cut into 1-by-2 1/2-inch rectangles. Transfer to the prepared baking sheets; prick with a fork and sprinkle with granulated sugar. Chill until firm, 15 minutes. Bake, switching the pans halfway through, until the cookies are firm, 13 to 16 minutes. Let cool completely on the baking sheets.
Put the chocolate in a microwave-safe bowl; microwave in 30-second intervals, stirring, until smooth; let cool. Add the remaining 1/2 stick butter; beat with a rubber spatula until incorporated. Mix in the remaining 1 tablespoon cocoa powder, the confectioners' sugar and the remaining 1/2 teaspoon vanilla until smooth. Sandwich between the cookies.