Preheat the oven to 180°C.
Grease and line a 20x20 cm cake tin.
Sift the flour, baking powder and salt together in a mixing bowl.
Beat the butter and sugar with an electric whisk until light and fluffy. Add the eggs one at a time, while continuing to beat. Stir in the vanilla.
Add the flour mixture and the milk alternately to the batter starting with the flour. Using a spatula blend in each addition (about 3-4 portions).
Move the batter into the cake tin, spread it out evenly.
Bake for 30 minutes until golden brown. Test it with a skewer - if it comes out clean, it's done.
Rest the cake for 5 minutes in the tin, then turn it out onto a wire rack. Let it cool completely, then cover it and chill it in the fridge for at least 2 hours (or even overnight) to firm up.
When you're ready to make the Lamingtons, cut the sponge cake into squares.
To make the icing: heat the cocoa powder, sugar, butter and milk in a heavy-based saucepan and stir until you get a smooth, silky, thick sauce.
Place the coconut in a shallow dish.
Using a fork, dip each square into the icing and coat on all sides. Then gently roll them in the coconut.
Rest them on a wire rack to dry.