Preheat conventional oven to 180°C (160°C for fan-forced). Line two baking trays with baking paper.
Stir together melted butter and egg in a medium bowl until combined and smooth.
Add cake mix and stir until it forms a soft dough. Cover with plastic and refrigerate for 30 minutes.
Roll dough into 30 balls and place on prepared baking trays, spacing them 6cm apart.
Bake until cracked at the surface and flattened (8-10mins).The cookies will be soft when ready. Do not over-bake. Transfer tray to a wire rack and let cookies cool completely on tray.
While the cookies cool make the icing by placing the frosting mix, butter and milk into a small mixing bowl. Using an electric mixer, beat on low speed for 30 seconds to combine. Increase speed to medium and beat for 3 minutes, scrapping down the sides of the bowl occasionally.
When the cookies are cooled sandwich together with the icing, pressing together gently to let the icing squeeze out the sides.
Place the crushed candy cane in a bowl and then roll edges of cookie sandwich in it, pressing so that the candy cane sticks to form a decorative edge.