01 - Preheat oven to 350°F. Grease an 8-inch square baking pan and line with parchment paper, leaving overhang on two sides for easy removal.
02 - In a medium bowl, whisk together the all-purpose flour, baking powder, and salt until evenly distributed. Set aside.
03 - Using an electric mixer, beat the softened butter and granulated sugar in a large bowl on medium-high speed until pale, light, and fluffy, approximately 3 to 4 minutes.
04 - Add the eggs one at a time, beating thoroughly after each addition until fully incorporated. Scrape down the bowl as needed, then mix in the vanilla extract.
05 - Gradually add the flour mixture to the butter mixture in three additions, alternating with the milk in two additions, beginning and ending with the dry ingredients. Mix until just combined, being careful not to overwork the batter.
06 - Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 20 to 25 minutes until a toothpick inserted into the center emerges clean and the top springs back when lightly pressed.
07 - Allow the cake to cool completely in the pan on a wire rack. Turn out the sponge, then wrap tightly in plastic wrap and freeze for 30 minutes to firm the crumb for easier handling during dipping.
08 - Combine the raspberries, water, granulated sugar, and lemon juice in a medium saucepan. Bring to a simmer over medium heat and cook for 5 to 7 minutes, stirring occasionally, until the sugar dissolves and the raspberries break down completely. Press the mixture through a fine mesh strainer to remove the seeds and discard the solids. Allow the syrup to cool to room temperature.
09 - Using a serrated knife, trim any domed top from the sponge to create an even surface. Cut the cake into 12 uniform squares.
10 - Spread the desiccated coconut in a wide, shallow dish. Working with one piece at a time, dip each sponge square briefly into the cooled raspberry syrup, allowing the excess to drip off. Transfer immediately to the coconut and roll gently, coating all sides evenly. Place each finished lamington on a wire rack.
11 - For filled lamingtons, slice each sponge square horizontally in half. Spread the bottom layer with raspberry jam, pipe or spoon whipped cream on top, then sandwich with the upper half before dipping in syrup and coating with coconut.
12 - Let the lamingtons rest on the wire rack for at least 15 minutes to allow the coating to set before serving.