Preheat oven to 350°F (180°C). In a bowl, sift together flour, cocoa, baking soda and salt. Set aside. In a measuring cup, stir together 3/4 cup (175 mL) milk and vanilla. Set aside.
Using an electric mixer (stand or hand-held), cream together 1/4 cup (60 mL) butter and the brown sugar until pale and fluffy, 3 to 5 min., then add egg. Reduce speed to low and alternately add portions of the dry ingredients and milk mixture, occasionally scraping down sides of bowl, finishing with dry ingredients. Beat just until a smooth batter forms.
Using a tablespoon, portion out level spoonfuls of batter and place mounds, spaced about 1 1/2 in. (4 cm) apart, on 2 parchment paper-lined baking sheets. Bake for 11 min., rotating baking sheets halfway through. Transfer pies to a wire rack and cool completely.
Meanwhile, make the filling. Using the mixer, whip remaining butter until fluffy, then add mint extract and remaining milk. With the mixer on low, gradually add icing sugar, occasionally scraping down sides of bowl. Beat until fluffy, about 4 min. Cover and set aside.
Fold 1/4 cup (60 mL) crushed candy canes into the filling. Sandwich level tablespoons of filling between 2 pies, spreading to the edges. At this point, they can be refrigerated overnight in a parchment paper-lined airtight container, with parchment between pies if they are stacked. Just before serving, dab filling edges into remaining candy cane and lightly pat pieces into place. Transfer to a platter and serve immediately.
Filling Switch-up: Add 1 tsp (5 mL) rum extract instead of mint extract to the filling and roll edges in 1⁄2 cup (125 mL) toasted shredded coconut instead of candy cane.