In a large bowl, beat the butter and sugars until light.
Add the vanilla, milk, coffee, and egg. Beat well.
Sift together the flour, cocoa powder, espresso powder, salt, baking soda, and baking powder.
Add gradually to the butter mixture, mixing together slowly. If you are using a stand mixer, make sure the bowl cover is on while you beat in the dry ingredients to avoid a chocolate mess.
When the dry ingredients are completely incorporated, roll half of the dough out at a time between pieces of parchment, lightly flouring the bottom piece of paper and turning the dough over several times to coat it before adding the top piece. Roll no thicker than 1/8-inch for crispy cookies. For softer cookies you can roll the dough out to 1/4-inch.
Place cookies at least 1/2-inch apart on parchment covered cookie sheets. Bake for 10-12 minutes. The baking time will depend on the thickness of your cookies and the desired crispiness. They should feel fairly firm when you remove them from the oven, but they will continue to firm up as they cool down. Allow the cookies to cool on the baking sheet.
Glaze
Whisk together the powdered sugar, vanilla, and as much coffee as you need to make a thick icing. Put the icing in a small plastic zipper bag with one corner snipped off and squeeze onto the cooled cookies.
Dairy Free
Use Dairy Free Margarine or coconut oil in place of butter, and milk substitute (almond, soy, etc)