In a large mixing bowl, whisk to combine the cocoa, flour, salt and baking powder. Set aside.
In a stand mixer fitted with the paddle attachment, beat the butter until it is smooth. Add the granulated sugar and beat again until the mixture is light and fluffy, about 3 minutes. Add the eggs and vanilla and beat again until combined. Add half of the cocoa-flour mixture, beat on low to combine, then add the rest and beat again until combined. Stop and scrape down the bowl as needed.
Turn the dough out onto the counter and divide in half. Wrap each half in plastic, shaping the dough into a flat disk. Chill for at least 1 hour and up to 2 days.
Roll out one disk of the cookie dough between two sheets of waxed paper or parchment paper. For chilled dough that is still somewhat hard, allow it to rest at room temperature for a few minutes. Tape down the waxed paper to anchor it, and roll from the center out, evenly, until the dough is 1/4- to 1/8-inch thick. Chill the rolled dough in the waxed paper layers for 15 minutes.
Arrange an oven rack in the middle position and preheat the oven to 350°F. Line 2 baking sheets with parchment paper.
Remove the rolled dough from the refrigerator and peel back and discard the top layer of paper. Dip cutters in flour (or cocoa powder) before cutting out shapes (dip each time you cut).
Transfer the shapes to the prepared baking sheets with a metal spatula. Bake for 8 minutes for 1/8-inch thick cookies and 11 minutes for 1/4-inch cookies, or until the edges are firm and the centers are slightly soft and puffed.
Press the dough scraps together and wrap in plastic. Chill and repeat the process, using fresh waxed paper each time you roll out a new batch.
Transfer baked cookies to a wire rack to cool to room temperature. Frost or sift with confectioners' sugar.