Preheat oven to 350ºF. In a large bowl, whisk together flour, cocoa powder, baking powder, baking soda, and salt. Set aside.
Beat butter and sugar in a standing mixer until light and fluffy, about 2 minutes, pausing occasionally to scrape down sides and bottom of bowl with a rubber spatula. Beat in eggs, one at a time, for about 30 seconds each, pausing to scrape down sides.
With mixer speed on low, beat in milk, sour cream, and vanilla extract, followed by the flour mixture.
Drop by level tablespoonfuls onto parchment-lined baking sheets. Bake 4 minutes, remove from the oven and sprinkle each with a bit of crushed candy cane, then bake 2 minutes more. Cookies are done when set but still a bit shiny (if you press the centre, it will leave an indent). They will look underdone - this is essential to chewy, delicious cakes.
To make filling:
Combine all ingredients in stand mixer with whisk attachment and beat until light and fluffy, about 5 full minutes.
To assemble:
Spread a heaping tablespoonful of frosting onto flat side of one cookie. Sandwich with another cookie, and transfer to a plate. Transfer to the fridge after you’ve frosted every half-dozen or so, as the filling is gooey and will start to spread out. Continue until all the cookies have been sandwiched. Keep refrigerated until ready to serve.