Preheat oven to 375 degrees. Grease two large cookie sheets, or line with parchment paper.
Whisk together the flour, cocoa powder, baking soda and salt.
In the bowl of a standing mixer, cream the butter and sugars together, 2-3 minutes. Add the egg and continue beating until fluffy, 1-2 minutes longer. Beat in milk and vanilla extract. Finally, beat in flour mixture to combine.
Drop by heaping tablespoons onto prepared cookie sheets (you should get 12-14 cookies).
Bake for 10-14 minutes, until top springs back when touched.
Cool completely. Spoon a heaping tablespoon of frosting on the flat half of one cookie, then top with another cookie. Repeat.