Preheat oven to 180C. In a large bowl sift or whisk together the flour, baking powder, and salt.
Beat the butter until soft. Add the sugar and beat until light and fluffy (about 2-3 minutes).
Add eggs, one at a time, beating well after each addition. Scrape down the sides of the bowl.
Add the vanilla extract and beat until combined. With the mixer on low speed, alternately add the flour mixture and milk, in three additions, beginning and ending with flour.
Spread the batter into the prepared pan and smooth the top with an offset spatula. Bake in preheated oven for about 25 - 30 minutes, or until a toothpick inserted in the centre of the cake comes out clean.
Cool the cake in its pan on a wire rack for 10 minutes. Then place a wire rack on top of the cake pan and invert, lifting off the pan. Re-invert.
Once the cake has completely cooled, cut into 16 two-inch (5 cm) squares. Wrap the cake in plastic wrap and refrigerate for several hours or even overnight (this is important).
Place the confectioners' sugar, cocoa powder, butter and milk in a heatproof bowl over a saucepan of simmering water. Stir the mixture until it becomes smooth and of pouring consistency.
To assemble Lamingtons:
Put the 16 squares of cakes on a wire rack that is placed over a baking sheet. Have the coconut ready on a large plate and the chocolate frosting.
Spoon or ladle the chocolate frosting over each square of cake, making sure you cover all sides. (It is best to do a few squares at a time).
With a small offset spatula or knife transfer the chocolate covered cake to the plate of coconut and roll the cake in the coconut, covering all sides.
Gently transfer the lamington to a clean wire rack to set. Repeat with the rest of the cake squares. Once the Lamingtons have set, store in an airtight container for several days.