Preheat the oven to 350° F. In a bowl, sift together flour, cocoa, baking soda, and salt. Set aside.
In a measuring cup, stir together 3/4 cup milk, and vanilla. Set aside. Using an electric mixer, cream together the brown sugar and 1/4 cup of the butter.
Add the egg and reduce speed to low. Add about a quarter of the combined dry ingredients and a third of milk mixture and beat together. Repeat until all ingredients have been thoroughly and a smooth batter forms.
Using a tablespoon, portion out mounds of batter, spaced about 2” apart, on parchment-lined cookie sheets.
Bake for 11 min. Transfer the baked shells to a wire rack to cool.
Using a mixer, whip together 1/2 cup of butter, until fluffy. Add mint extract and 2 tablespoons milk. With the mixer on low, gradually add powdered sugar. Beat until fluffy. Fold 1/4 cup crushed candy canes into the filling.
Using level tablespoons, top half of the shells with icing, then add a second shell to create a sandwich. Be sure to spread the filling to the edges of each.
Roll the edges of each whoopie pie in the remaining crushed candy canes, and serve.