In a food processor mix together sugar, brown sugar, flour, cocoa, baking soda and salt. Add the butter a few pieces at a time, with the motor running. Pulse to incorporate. Add the egg and vanilla. Pulse until the dough starts to form. Turn the dough onto a clean surface. Form into a log/rope that is about 1 1/2 inches in diameter. Wrap in plastic wrap, and let chill in the refrigerator for 60 minutes.
Preheat oven to 350° F.
Remove dough from the fridge and remove plastic wrap. Using a sharp knife, slice dough about 1/4 inch thick. Place dough on a lined baking sheet about 2 inches apart. Bake for 12-15 minutes until set and cookies look dry. Remove from the oven and cool completely.
Meanwhile, prepare the filling.
To prepare the filling beat together butter, vanilla, peppermint and salt until fluffy. Add sugar and beat until fully incorporated. Add milk 1 Tbs at a time and beat until the desired consistency.
Spoon or pipe filling into the center of a cookie. Top with another cookie, pressing down gently to spread the filling to the edge.