Grate butter on box grater and put into freezer until ready to use.
Preheat oven to 400 degrees. Line a cookie sheet with parchment paper or silicone baking mat.
Using your food processor, or pastry hand mixer, combine flour, cocoa, baking powder, and salt. Pulse or mix in the butter just until the butter is dispersed evenly through the flour and pea size.
Mix together the egg, milk, coffee powder, and peppermint extract. Add into flour mixture. Mix until dough just comes together.
Turn dough out onto floured surface and pat into 8 inch diameter circle. Cut into 8 wedges and place on prepared baking sheet.
Bake for 20- 25 minutes.
Remove scones from oven and cool slightly.
Crush candy canes into a powder.
Melt white chocolate over double boiler or in microwave, stirring after 30 second intervals.
Drizzle chocolate over scones and then sprinkle with candy cane dust.