Preheat oven to 350 degrees F (180 degrees C) and place oven rack to middle position.  Butter, or spray with a nonstick cooking spray, the bottom and sides of an 8 inch (20 cm) square cake pan.  Set aside.

In a large bowl sift or whisk together the flour, baking powder, and salt.  Set aside.

In bowl of electric mixer, 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 a preheated oven for approximately 25-30 minutes, or until a toothpick inserted in the center of the cake comes out clean.

Cool the cake in its pan on a wire rack for 10 minutes.  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 it into 16 two-inch (5 cm) squares.  Wrap the cake in plastic wrap and refrigerate for several hours or even overnight.  The reason for doing this is that it is much easier (less crumbs) to coat a cold cake with frosting.

Chocolate Frosting:  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:  Make a production line; put the squares of cakes on a wire rack that is placed over a baking sheet (to catch the drips).  Have ready the coconut 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 of cake 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.

Note:  When you ladle the frosting over the cake, some of the frosting will drip onto the baking pan.  Pour this frosting back in your bowl and reuse (strain if necessary).  If the icing becomes too thick to pour, simply place the frosting back over the saucepan of simmering water and reheat until it is of pouring consistency. (You may have to do this a few times as the frosting has a tendency to thicken over time.  Add a little more milk to frosting if necessary to get the right consistency.) 

Makes 16 2-inch (5 cm) squares.