1. To make the spongecake, butter a 9-inch (23 cm) square cake pan and line the bottom with parchment paper. Preheat the oven to 350ºF (180ºC).
2. In the bowl of a stand mixer, whip the eggs and granulated sugar and salt on high speed for five to ten minutes, until thick and the batter forms a well-defined ribbon that remains on top of the batter when you lift the whip. Stir in the vanilla.
3. Fold the flour into the egg mixture by putting the flour in a sifter or mesh strainer and sifting the flour over the top of the beaten eggs while simultaneously folding the flour in using a whisk.
(You might want to steady the bowl on a damp towel, twisted and rolled up into a circle, or get someone to hold the bowl while you fold.) Fold in the melted butter until no streaks of butter are visible, but do not overfold.
4. Scrape the batter into the prepared pan and bake for 30 minutes, or until the cake begins to pull away from the sides of the pan. Remove from oven and let cool completely.
5. When cool, unmold the spongecake onto a cutting board and remove the parchment paper. Trim the ends and cut the cake in half horizontally using a serrated bread knife. (I find it easier to cut the cake into two rectangles first, and cut each one separately.)
6. Make the chocolate icing by melting together the chocolate, butter, and milk in a heatproof bowl set over a pan of simmering water. Remove the bowl from the pan of simmering water when smooth, then whisk in the powdered sugar and cocoa powder.
7. Spread a generous 1/2 cup (125 ml) of the chocolate icing over one layer of the spongecake, then top with the other half of the spongecake, sandwiching the two together with chocolate icing in the middle.
8. Cut the cakes into sixteen squares and whisk two tablespoons of boiling water into the icing. Put the coconut into a shallow baking dish or bowl.
9. Use your hands to dip the Lamingtons into the chocolate, rolling them around to make sure each side is coated with the chocolate icing, then wipe off any excess on the side of the bowl. Place the Lamingtons in the vessel of coconut, tossing them around gently to get them coated on all sides. (I do two at a time.)
10. Once iced and tossed in coconut, place the Lamingtons on a wire cooling rack and let stand until the icing firms up a bit.