Place self-rising flour, superfine sugar, and 1/2 cup butter in a food processor; pulse until the mixture resembles coarse crumbs, about 20 seconds.
Place eggs, milk, 1/4 cup lemon juice, and 2 teaspoons lemon zest in the food processor, and pulse until batter is just mixed, about 10 seconds more.
Transfer batter to a large bowl; fold 1 cup coconut into the batter.
Pour batter into prepared baking pan, smoothing the surface of the batter.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 1 hour. Cool in the pan for 10 minutes before removing to cool completely on a wire rack.
Beat 1/4 cup butter with an electric mixer in a bowl until light and creamy.
Gradually beat in confectioners' sugar, coconut milk powder, 2 tablespoons lemon juice, and 2 teaspoons lemon zest until frosting is smooth.
Spread frosting over completely cooked cake.
Sprinkle remaining 1/4 cup coconut atop frosted cake.
