Line an 8 inch square pan with non-stick baking paper. Preheat the oven to 175 C.
In a mixing bowl, beat the butter and sugar with a hand mixer on medium speed till combined. It will look a little gritty and crumbly because the amount of butter is small compared to a regular cake batter. Add the vanilla and eggs and beat again for a 1 to 2 minutes until pale and slightly thickened.
Sift in the flour, baking powder and cornflour. Fold to combine, then gradually add the milk, stirring as you go, till you have a smooth and thick batter. If needed, run a hand mixer through it to remove any lumps but do not overmix the batter.
Pour the batter into the prepared pan and bake for 20 to 25 minutes until the top is golden-brown and a toothpick poked into the center of the cake comes out clean. If the top is still a little pale but the cake is cooked through, that's ok.
Allow the cake to cool for an hour at room temperature, then cut into 24 pieces and chill in the fridge for another hour (the photos above show 16 pieces, but after making these recently, I recommend 24 pieces for easier dipping and eating). If you're not planning to assemble the lamingtons immediately, you can let the cake cool completely, then cover and leave it at room temperature overnight, then chill the pieces for about 30 to 40 minutes before assembly. Chilling makes it easier to dip the pieces in the icing.
When ready to assemble, make the icing. Sift the sugar and cocoa together in a bowl, then add the chopped chocolate. Pour the boiling water in and stir or whisk slowly till you have a smooth sauce. It should be easy to dip into but also not be too runny to coat the cake properly.
Spread about half of the desiccated coconut out on a large platter. Dip each chilled piece of cake into the icing, turning it over with a fork to coat it completely, then lift it out, let the excess drip off and place it on the coconut platter, rolling and turning with your fingers to make sure all sides are covered in coconut (some of the chocolate will end up on the plate, don't worry about it). Transfer to a serving platter or airtight tin. Repeat with the rest of the cake, replenishing the coconut as needed. For a jam variation, see notes.
Chill the prepared lamingtons for an hour, then dig in! Store in an airtight tin in the fridge for a week and let them soften at room temperature for about 15 minutes before enjoying. Happy eating!