Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a mixing bowl, beat the butter and sugar with an electric mixer until light and fluffy.
Add the egg and vanilla extract, mixing until fully combined.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until a dough forms.
Roll out the dough on a lightly floured surface to about 1/8-inch thickness and cut into circles using a cookie cutter.
Place cookies on the prepared baking sheet and bake for 8-10 minutes, until set but not overly crisp.
Let cookies cool completely on a wire rack before filling.
For the filling, beat together the powdered sugar, softened butter, milk, and vanilla extract until smooth and creamy.
Spread or pipe the filling onto one cookie, then sandwich with another cookie.
Chill for 10 minutes to set before serving.