Unwrap 1 of the candy canes and place in a food processor. Pulse until the candy cane is broken up into small pieces. Pour the pieces onto a plate.
Place coconut oil and white chocolate chips in a wide but shallow dish. Cook in the microwave for 15 seconds then with a fork whisk to combine ingredients. If the chocolate chips are not melted fully cook for an addition 15 minutes. Dip the tops of two mugs in the melted chocolate. While the chocolate is still warm dip the mugs into the candy cane crumbs to coat the rims of the mug. Place the mugs in the refrigerator.