Preheat oven to 350F and line two baking sheets with parchment paper.
In a medium bowl, whisk together the flour, cocoa powder, baking soda and salt. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar until light and fluffy. Add egg and vanilla. Beat until well combined. Gradually add in the milk, scraping the bottom of the bowl as needed. The batter might curdle, but don’t worry, it will come together when you add the dry ingredients.
Slowly add dry ingredients. Mix until just combined.
Using a 1-oz cookie or ice cream scoop, place dough onto a parchment-lined baking pan, 12 per pan. Bake for 10-12 minutes or until a toothpick inserted into the center comes out clean. Let cool completely before frosting.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter until smooth and creamy. Add in the vanilla, one cup of powdered sugar, and one tbsp of milk. Mix well. Add the remaining powdered sugar, one cup at a time, alternating with 1 tbsp of milk at a time. Add in the last tbsp of milk if needed for the desired consistency.
Transfer buttercream to a disposable pastry bag and snip the end. When cookies have cooled completely, pipe 2 tablespoons buttercream onto the flat side of half the cookies. Sandwich with remaining cookies, pressing down slightly so that the filling spreads to the edge of the cookies.
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