Preheat the oven to 350°F (180°C). Lightly butter a 9-by-13-inch (23-by-33-cm) baking pan and line with parchment paper long enough to hang over the 2 long sides.
In the bowl of a stand mixer on medium speed, combine the butter, coconut oil, and superfine sugar, and beat until light and fluffy, 5 to 7 minutes.
Meanwhile, in a separate large bowl, mix together the flour, baking powder and salt.
Add the vanilla and coconut extracts to the butter mixture and mix briefly to combine. Then add the egg whites, 1 at a time, mixing until completely combined before adding another.
Add the flour mixture in 3 additions, alternating with the coconut milk, starting and finishing with the flour. Fold in the desiccated or shredded coconut.
Scrape the cake batter into the prepared pan and spread it evenly. Bake until the cake springs back to a light touch or a skewer inserted in the middle of the cake comes out clean, 27 to 35 minutes.
Let the cake cool in the pan for about 15 minutes. Use the parchment paper to lift it out of the pan and place it on a wire rack to cool completely.