Preheat the oven to 180°C (350°F/Gas 4). Lightly grease a shallow 23cm (9inch) square cake tin and line the base and sides with baking paper.
To make the cake, sift the flour and cornflour into a large bowl. Add the butter, sugar, vanilla, egg and milk. Using electric beaters, beat on low speed for 1 minute, or until the ingredients are just moistened. Increase the speed to high and beat for 3 minutes, or until free of lumps and increased in volume. Pour into the tin and smooth the surface. Bake for 50–55 minutes, or until a skewer inserted into the centre of the cake comes out clean. Leave the cake in the tin for 3 minutes before turning out onto a wire rack to cool.
Using a serrated knife, trim the top of the cake until flat. Trim the crusts from the sides, then cut the cake in half horizontally. Using electric beaters, beat the cream in a small bowl until stiff peaks form. Place the first layer of cake on a board and spread it evenly with cream. Place the remaining cake layer on top. Cut the cake into 16 squares.
To make the icing, sift the icing sugar and cocoa into a heatproof bowl and add the butter and milk. Sit the bowl over a saucepan of simmering water, making sure the base of the bowl does not touch the water, and stir until the icing is smooth and glossy, then remove from the heat.
Place 90g (3¼oz/1cup) of the coconut on a sheet of baking paper. Using two spoons, dip a piece of cake in chocolate icing, then hold the cake over a bowl and allow the excess to drain. (Add 1tablespoon boiling water to the icing if it seems too thick.) Roll the cake in the coconut, then place on a wire rack. Repeat with the remaining cake, adding extra coconut for rolling as needed.