Preheat oven to 375 degrees Fahrenheit and set oven racks to the middle of the oven.
In a large bowl, mix together the flour, cocoa, baking soda, baking powder, salt, and sugar.
While mixing on low speed, add the butter and egg, and mix until the dough forms into a giant clump.
Line a baking pan with parchment paper or grease well.
Take a teaspoon of batter, and roll it into a ball with your hands. Then lightly press down on the ball to form a patty (not entirely, these will flatten out a bit in the oven).
Place about 2 inches apart on the baking sheet.
Bake for 8-9 minutes and then cool on a wire rack.
To make the icing, cream butter and shortening in a bowl.
Then add the confectioners' sugar and vanilla. Keep mixing until the batter is light and fluffy.
To assemble cookies, take a spoon and put a little glob of frosting in the middle of the cookie.
Push down with the top cookie to make the icing cover all ends of the cookie.
You can also use a pastry bag to pipe the frosting if you would like.
Makes about 25-30 sandwich cookies.