In a medium bowl, whisk together the flour, cocoa powder, baking soda, and salt. Set aside.
In a large bowl, beat the butter and powdered sugar until smooth. Add the egg and vanilla and mix until combined.
Gradually mix the dry ingredients into the wet ingredients until a thick dough forms.
Flatten the dough into a disc and refrigerate for 20–30 minutes for easier rolling and cleaner cuts. (optional)
Preheat the oven to 350°F. Roll the dough between parchment paper to about ⅛-inch thick. Cut into rounds and place on a parchment-lined baking sheet.
Bake for 8–10 minutes, until the cookies are set. Allow cookies to cool completely.
Beat the butter until smooth. Add powdered sugar, vanilla, and milk, mixing until light and fluffy.
Spread or pipe filling onto one cookie and sandwich with another cookie.