In a standing mixer with a paddle attachment, cream butter and brown sugar on medium speed for 2 minutes. Add in the egg, vanilla extract, and salt and beat 1 minute more.
Turn mixer to low and mix in the cocoa, flour, and baking soda, mixing until just combined.
Use a 1/2 oz cookie scoop (or use a 1 oz cookie scoop and divide the dough in half) to scoop balls of dough. Roll each ball between palms to form a perfect circle, then place on prepared baking sheet and push down slightly, making a slightly flattened disk. You should have 32 scoops of dough.
Bake for 7 minutes, then remove from heat and allow cookies to cool on baking sheets until they reach room temperature.
To make the buttercream, beat the butter, sugar, peppermint extract and milk until light and fluffy, about 2 minutes. Gently fold in the candy cane pieces until just combined. Place frosting into a piping bag with a medium tip.
Pipe roughly 1 tablespoon of frosting onto half of the cookies, then sandwich with remaining cookies. Drizzle with chocolate and top with crushed up candy canes! Keep refrigerated. Will last up to 5 days in the fridge.