Preheat the oven and prep the muffin tin. Preheat the oven to 350°F and line 12 muffin tins with paper liners.
Mix the cupcakes. In a large bowl, whisk together the milk, vegetable oil, eggs, vanilla, and peppermint extract. Add the granulated sugar and whisk to combine. Add the flour, cocoa, and baking powder. Mix just until combined.
Fill the muffin tins and bake. Divide the batter between the cupcake tins, filling each one ¾ of the way full. Bake the cupcakes for 18-20 minutes, or until a toothpick inserted into the center comes out clean.
Cool. Remove the cupcakes to a wire rack and set them aside to cool before frosting.
Blend the buttercream icing. Add the butter to the bowl of a stand mixer fitted with the paddle attachment. With the mixer on low speed, slowly add the powdered sugar. Place the white chocolate in a microwave safe dish and heat in 15-second increments, stirring between each, until fully melted. Once all of the powdered sugar has been incorporated, add the melted white chocolate and vanilla extract. Turn the mixer to high speed and beat until the buttercream is light and fluffy - about 2 minutes. If the buttercream is too dry, add a tablespoon of milk to bring it together.
Frost.Transfer the buttercream to a piping bag fitted with an open star tip or your favorite tip. Frost the cupcakes as desired and top them with crushed candy canes.
Store. Store the cupcakes in an airtight container in the fridge for up to three days.