In a bowl, mix together sugars and butter. Then beat in eggs, vanilla extract, and peppermint extract. until combined.
Mix in cocoa powder, baking soda, salt, and flour. Then stir in chopped chocolate until combined.
Using a 1 Tablespoon cookie scoop, place cookie dough on a parchment-lined cookie sheet. Flatten each cookie slightly. Bake for 5 minutes and then remove the cookie sheet and firmly tap on the counter. Return the cookie sheet to the oven and bake an additional 2 minutes.
Let cool on the cookie sheet for 2 minutes before allowing them to cool completely on a cooling rack.
Make the peppermint buttercream by beating with a mixer one stick of butter for 2 minutes before beating in 1/2 cup of powdered sugar, milk, and peppermint extract. Gradually add in an additional 1 1/2 cups of powdered sugar. Beat until combined.
Once the cookies are cooled, spread peppermint buttercream on the bottom of half the cookies (I used a pastry bag and tip, but you could just spread with a butter knife). Then top each cookie with an additional cookie.