Make the Cake: Preheat oven to 375 degrees F. Line two baking sheets with parchment paper.
Whisk together the flour, cocoa powder, baking soda, and salt in a medium bowl. In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter, shortening and brown sugar on low speed until just combined. Increase the speed to medium and beat until fluffy and smooth, about 3 minutes. Add the egg and vanilla and beat for another 2 minutes.
Add half of the flour mixture and half of the milk to the batter and beat on low until just incorporated. Scrape down the sides of the bowl. Add the remaining flour mixture and ½ cup milk and beat until completely combined.
Drop about 1 tablespoon of batter onto one of the prepared baking sheets and repeat, spacing them at least 3 inches apart. Bake one sheet at a time for about 10 minutes each, or until the pies spring back when pressed gently. Remove from the oven and let the cakes cool on the sheet for about 5 minutes before transferring them to a rack to cool completely.
Make the Filling: In the bowl of a stand mixer fitted with the paddle attachment, beat together the powdered sugar and the butter, starting on low and increasing to medium speed, until the mixture is crumbly, about 1 minute. Add the heavy cream, vanilla, peppermint extract, and salt and beat on high until smooth, about 3 minutes.
Assemble the Whoopie Pies: Spread filling onto the flat side of one cake using a knife or spoon. (Alternatively, you can use a pastry bag with a round tip to pipe the filling onto the cakes, which will give you a smoother, neater presentation.) Top it with another cake, flat-side down. Repeat wit the rest of the cakes and filling. Roll the edges in crushed candy canes.
Whoopie pies are best eaten within a day of making them. Store in a wide, shallow container arranged in a single layer for up to 3 days. You can also wrap them individually in plastic wrap.