Preheat oven to 350°F. Line two standard muffin tins with 24 decorative paper liners.
In the bowl of a stand mixer fitted with the paddle, mix sugar, flour, cocoa, baking powder, baking soda and salt on low speed until well combined.
Add eggs, milk, oil and vanilla extract. Mix on medium speed until batter is smooth. Pour in boiling water and carefully mix on low until just incorporated.
Divide batter among muffin cups. Bake cupcakes for about 18 minutes, until a tester comes out clean. Cool completely on a wire rack before frosting.
Make the frosting:
In the bowl of a stand mixer fitted with the whisk (or in a large bowl with electric beaters), beat the butter for three to four minutes until light and creamy. Sift together the powdered sugar and cocoa powder, then add to mixer (with mixer off) along with milk, vanilla and peppermint. Beat on low until incorporated, then on high for three to four minutes until smooth and fluffy. Let sit for 10 minutes to help powdered sugar chillax a bit into the mix. (Technical term.) If frosting is too thick, beat in up to two tablespoons more milk a little at a time.
Scoop frosting into a large piping bag fitted with a large star tip and pipe some frosting straight down onto each cupcake. Dust each cupcake with some crushed candy cane and spear with a mini candy cane, making sure it penetrates the cupcake and not just the icing so it doesn't fall out. That's it!