In a bowl, mix 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 cookie scoop, place cookie dough on a parchment-lined cookie sheet. Flatten each cookie slightly.
Bake for 5 minutes, remove the cookie sheet, and firmly tap on the counter. Return the cookie sheet to the oven and bake an additional 2 minutes.
Let the cookies 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 or pipe the peppermint buttercream on the top of the cookies (I used a pastry bag and tip, but you could just spread with a butter knife). Then top each cookie with crusted peppermint candy (or candy cane).