To make the frosting: In a heatproof mixing bowl, add the white chocolate and heavy whipping cream. Microwave for 30 second intervals, stirring between each one to prevent burning, until completely melted.
Add in the peppermint extract and stir to combine. Cover with plastic wrap and chill for at least 2 hours.
Preheat the oven to 350°F. Line a 9-inch baking pan with parchment paper.
To make the batter: In the bowl of a stand mixer fitted with the paddle attachment (or using a hand mixer), add the cane sugar, cake flour, cocoa, baking powder, and sea salt. Whisk together to combine.
Add in the canola oil, milk, eggs, and vanilla extract and mix on low until combined into a smooth batter. Transfer to the prepared baking pan and spread out evenly.
Bake for 25 to 30 minutes or until a wooden pick inserted in the center comes out clean. Allow to cool completely in the cake pan.
Remove the white chocolate ganache from the fridge and whisk, using a hand mixer, until it's light and fluffy, about 2 minutes.
Remove the cake from the pan.
Top the cake with the frosting and spread evenly. Top with crushed candy canes.
Store in an airtight container for up to 5 days.