Line an 8 inch square pan with non-stick baking paper or lightly grease the pan. 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 well. This can be done entirely by hand.
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.
Pour the cake into the prepared pan and bake for 25 to 30 minutes until the top is golden-brown and a toothpick poked into the center of the cake comes out clean.
Allow the cake to cool for an hour at room temperature, then cut into squares and chill in the fridge for another hour. Alternatively, you can make the cake the night before and let it cool at room temperature overnight.
Once the cake has cooled, make the icing. Combine the sugar, cocoa powder and chocolate in a bowl, then pour the boiling water over the mixture and whisk vigorously till you have a smooth sauce and the chocolate has completely melted. It should be easy to dip into, but not super runny. If needed, add a couple of extra tablespoons of hot water.
Spread about half of the desiccated coconut out on a large platter. Dip each cooled piece of sponge into the icing, turning it over with a fork to coat it completely, then lift it out 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.
Chill the prepared lamingtons for an hour, then dig in! Store in an airtight tin in the fridge for a week. Happy eating!