For the cookies: Whisk together the flour, cocoa, baking soda, baking powder and salt in a large bowl. Sift into another large bowl.
Beat the granulated sugar and butter together in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 5 minutes. Add the egg and beat until incorporated. Beat in the vanilla. Alternate adding the dry mixture and the milk, beginning and ending with the dry mixture. Mix until just incorporated. Refrigerate the dough until chilled, about 1 hour.
Adjust the oven racks to the top and bottom thirds of the oven and preheat to 400 degrees F. Line 2 baking sheets with parchment and lightly coat with cooking spray.
Drop tablespoonfuls of batter about 3 inches apart on the prepared baking sheets (about 6 cookies per baking sheet). Keep the remaining batter chilled while working.
Bake until the cookies puff up and the tops are no longer glossy, about 7 minutes. Let cool for a few minutes on the baking sheets, then transfer to a wire rack to cool completely, about 1 hour. Let the baking sheets cool, then reline with parchment, lightly coat with cooking spray and bake the remaining batter in batches (cool, reline and spray the baking sheets each time).
For the filling: Beat the confectioners' sugar and shortening in a large bowl with an electric mixer on medium speed until fluffy. Add the marshmallow creme, milk and peppermint extract and beat until light and airy, about 5 minutes.
Spread a heaping tablespoon of the filling on half of the cookies. Sandwich with the remaining halves, pushing down slightly to move the filling to the edge. Put the crushed candy canes on a plate and roll the sides of the whoopie pies in the candy.
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