First, brown the butter. In a small saucepan over medium-low heat, melt all of the butter and stir constantly until melted and simmering. Continue to simmer, on medium heat, for at least a few minutes, until the popping and hissing sounds have completely subsided (this is the water in the butter burning off - you'll know once it's silent). Continue to cook, stirring constantly, until the butter reaches a deep amber color and smells nutty or caramelly. Once it reaches that color, immediately take off the burner and let sit until cool. Once cool, make the cake.
Preheat oven to 350° Fahrenheit. Lightly grease a 9x13 inch pan and line with parchment paper, grease the paper and set aside. In a large bowl, using an electric mixer, beat together the room temp brown butter with the sugar sugar until lighter and fluffy, then add the sour cream, eggs, vanilla, and milk and beat until smooth. Finally, add in the flours, starches, baking powder and soda, and salt and whisk until fully combined.
Pour batter into prepared pan and bake for 30-40 minutes or until a butter knife inserted in to the middle of the cake comes out clean. Let cool completely on a wire rack, and then wrap entire cake in plastic wrap and plate a few large books on top to carefully press the cake (so that your lamingtons are perfect cubes). Leave the cake to press for 6+ hours (or overnight).
Once the cake is pressed, unwrap and slice into 24 cubes about 2x2 inches. I like to trim each one to create a perfect cube, but if you're not picky, just leave as is! Once your cubes are cubed, get the jam in the middle (totally optional, lamingtons are great without them middle layer of jam too!). Slice them in half as if you were slicing bread for a sandwich, then spread about a teaspoon of jam on one half and then make your lamington sandwich. Place all jammed lamingtons into a tray and freeze for 1 hour to set the jam/cake before dipping in the glaze.
While the lamingtons are freezing, make the toasted coconut and chocolate glaze. First, preheat oven to 325° Fahrenheit. Line a cookie sheet with parchment paper and then pour the three cups of desiccated coconut evenly on the tray. Bake for a few minutes only, stirring/moving the coconut around frequently to avoid burning. Once it's toasted to your liking, remove and let cool, then put into a bowl for dunking. Then make the glaze.
Put an inch of water into a small saucepan and bring to a simmer over low heat. Place a metal bowl on top of the pan so that the bowl is not touching the water. Stir the chocolate, butter, salt, milk and cream together in the bowl as it melts. Once completely melted and combined, take the bowl off of the pan (turn off heat) and whisk in the powdered sugar and cocoa powders.
Remove lamingtons from freezer and use forks or a skewer to dunk them in the warm chocolate glaze until completely coated. I let them drip on a wire rack for a few seconds before dunking the cube in the toasted coconut. Dunk all lamingtons in glaze then coconut and place back on a wire rack then freeze lamingtons again for about 30-40 minutes just to set the chocolate. Then eat! I love them still chilled, but you can bring them completely to room temp before serving too.