Preheat the oven to 180°C/Fan°C/Gas 4. Whisk together the butter and caster sugar until light and creamy. Add the flour and cocoa and beat together to make a soft dough. If the mixture is too crumbly, add a little milk.
On a flour-dusted surface, roll out the dough to 5 mm thick and then cut out fingers of cookie dough about 12 x 3 cm in size. Using a spatula, carefully lift the cookie shapes onto the baking sheets. Make bark patterns in the dough by scraping it gently with the tines of a fork.
Bake in the preheated oven for 10–12 minutes until the cookies are just firm. Leave to cool on the baking sheets for 5 minutes, then move to a wire rack to cool completely.
Melt the white chocolate in a heatproof bowl resting over a pan of simmering water. Stir in the peppermint extract. Dip the end of each cooled cookie in the melted chocolate and then sprinkle each with the crushed candy canes. Place on a sheet of baking parchment and leave in a cool place to set. The cookies will store for up to 5 days in an airtight container. It is best to store the cookies flat, in single layers, and between sheets of baking parchment.