Preheat the oven to 350 degrees F (180 degrees C). Line the baking tray(s) with baking paper.
Cream the butter and sugar in a small bowl with a wooden spoon until light and fluffy.
Add the sifted flours and cocoa alternately with the milk while mixing with a butter knife until a soft dough forms.
Turn the dough out onto a lightly floured surface and roll it into two equal-sized balls. Place a ball in between two sheets of baking paper and roll as thin as you can, about 1/8-inch (2mm) thick.
Using a 1 1/2-inch (4cm) cookie cutter, cut a batch of cookies and place them on the prepared baking tray. Repeat with the second ball of dough.
Place the tray(s) with the chocolate cookies in the preheated oven and bake for 10 minutes.
Peppermint Cream
In a small bowl add the egg white and 2 tablespoons of confectioner's sugar, then whisk with a fork. Keep adding the confectioner's sugar 2 tablespoons at a time.
Add the peppermint extract to the peppermint cream.
Chocolate Coating
In a heatproof bowl over simmering water add the dark chocolate and coconut oil.
Allow to slowly heat and melt stirring occasionally until fully melted. Remove from the heat as soon as it has melted so it doesn’t overheat and ‘burn’.
To Assemble
Make a peppermint sandwich between two chocolate cookies. A teaspoon can be used to help get the peppermint in between the cookies.
Using a fork, dip the filled cookie into the chocolate and drain off any additional chocolate. Place on a baking paper-lined tray to set, about 1 hour.