Preheat oven to 350°F. Line and grease a baking sheet with a silicone baking mat.
In a bowl, mix together all ingredients for the cake and bake for about 15 minutes until a toothpick comes out clean. Run a knife around the edge to help release cake freely.
While the cake is still warm, moisten a towel, and sprinkle finished cake with cocoa powder. Flip cake over onto towel, sprinkle the top with more cocoa powder, and carefully roll with towel starting with the widest edge. Let cool completely. (The damp towel helps the cake roll more easily.)
For the filling, combine all ingredients except crushed candy cane, and whip until blended.
For the ganache, place all ingredients in a heat-proof bowl, pop into the microwave and heat in 30 second intervals until melted.
Unroll cooled cake and fill with buttercream filling. Sprinkle with crushed peppermints, then re-roll carefully, tidying up the ends as you go (some filling may escape the ends, so you may have to trim the sides for a nice clean roll).
Place on a cooling rack. Drizzle with ganache and top with more crushed candies and chocolate shavings.