Preheat your oven to 375 degrees F. Line two baking sheets with parchment paper.
In a medium bowl, whisk together all-purpose flour, unsweetened cocoa powder, baking soda, and salt.
In the bowl of a stand mixer, combine unsalted butter, vegetable shortening, and dark brown sugar. Beat on low speed until just combined, then increase to medium and beat until fluffy and smooth, about 3 minutes.
Add the egg and vanilla extract to the mixture, beating for another 2 minutes.
Gradually mix in half of the dry ingredients along with half of the milk. Beat on low until just incorporated, then scrape down the sides of the bowl.
Add the remaining dry ingredients and ½ cup of milk, mixing until completely combined.
Using a tablespoon or a cookie scoop, drop about 1 tablespoon of batter onto the prepared baking sheets, spacing them at least 3 inches apart.
Bake one sheet at a time for about 10 minutes, or until the pies spring back when lightly pressed.
Remove from the oven and let the cakes cool on the sheet for about 5 minutes before transferring them to a wire rack to cool completely.
In the meantime, prepare the filling by combining powdered sugar and unsalted butter, starting on low and increasing to medium speed.
Add heavy cream, peppermint extract, and salt. Beat on high until smooth.
To assemble, spread the filling onto the flat side of one cake using a knife or spoon. Top with another cake, flat side down.
Roll the edges in crushed candy canes for a festive touch.
Store leftovers in a wide, shallow container arranged in a single layer for up to 3 days.
