About 15-20 minutes before you are ready to assemble, prepare the white chocolate icing: Boil water in a small saucepan over medium heat. Whisk together the milk and chopped chocolates in a small, heat-safe bowl. Rest the bowl on top of the boiling water (making sure the bowl doesn't touch the water). Whisk constantly until the chocolate is fully melted and the mixture is smooth. Remove from heat- careful- the bowl will be hot.
Add the sifted powdered sugar to the bowl with the melted chocolate one cup at a time. Whisk until smooth before adding the next cup. Set aside.
Place a wire cooling rack over a rimmed baking sheet. This is where you will put your sandwiches after they have been dipped in chocolate. Set aside.
Place about a cup of the coconut in a shallow bowl. (Replenish with more coconut as you go.) Set aside.
Take a few of the frozen sandwiches out of the freezer. Gently stick the side of one with a fork and place over the bowl with the melted chocolate. Use a spoon to spoon the white chocolate icing over the sandwich until all sides are completely coated. Let it drain over the bowl to catch the drippings for a few minutes. Transfer to the wire rack. Repeat with a few more sandwiches (I did about 4-5 at a time).
Take a sandwich that has already been dipped in white chocolate and rest it on a fork. Place over a bowl with the coconut and use your hand to cover every side with coconut, pressing slighting so it sticks. If your layers start slipping around, make sure to straighten them and cover every inch with coconut. Set aside and repeat with remaining sandwiches.
Follow Step 5 and Step 6 for the remaining sandwiches in the freezer. I found it was best to work in small batches (about 4-5 at a time) so they do not warm up too much.
Place the finished lamingtons in the refrigerator for about 15 minutes to set. Take the cakes out of the refrigerator a few minutes before serving to come slightly back to room temperature. Enjoy!