Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a bowl, whisk together flour, cocoa powder, baking soda, and salt.
In another bowl, beat butter and sugar until fluffy. Add egg and vanilla, beat until combined.
Alternate adding dry ingredients and buttermilk to the butter mixture, starting and ending with dry ingredients.
Scoop batter onto baking sheets, spacing 2 inches apart. Bake for 10 minutes. Cool completely.
For filling, beat butter and peanut butter until smooth. Gradually add powdered sugar, then milk, beating until fluffy.
Spread filling on flat side of one cookie, top with another cookie. Repeat with remaining cookies.