Preheat the oven to 350 degrees F.
In the bowl of a stand mixer, cream the butter and sugar together.
Add the egg, vanilla, and milk and combine.
In a separate bowl, combine the flour, baking powder, baking soda, salt, and cocoa powder.
Add the dry ingredients into the wet ingredients and mix until combined.
Use a 1 tablespoon cookie scoop to scoop the cookie dough out onto a baking sheet lined with parchment paper. Roll each cookie into a smooth ball.
Spray the back of a measuring cup with grease and use it to gently flatten each cookie dough ball into a disc.
Refrigerate the dough for 20 minutes, then bake for 6 minutes.
Prepare the icing while the cookies cool.
Beat the butter in the bowl of a stand mixer.
Add the powdered sugar, vanilla, and milk. Beat until fluffy.
Scoop a heaping tablespoon of the icing on one side of the cookie and sandwich it together with another cookie.