Melt the butter in a small bowl, add 1 tablespoon of flour and stir together until it is smooth paste. With a pastry brush thoroughly coat the bottoms and sides of two 9-inch aluminum cake pans and line them with parchment paper.

Crack the three eggs into a bowl large enough to hold all three eggs.

In the bowl of a stand mixer fitted with a paddle attachment, combine sugar, baking powder, salt, vanilla extract, butter, and virgin coconut oil. Mix on low speed to roughly incorporate, then increase to medium and beat until fluffy and light, about 5 minutes. Halfway through, pause to scrape the bowl and beater with a flexible spatula.

Cool cakes in their pans for 15 minutes, then run a butter knife around the edges to loosen. Invert onto a wire rack, peel off parchment, and return cakes right side up. Allow the cakes to completely cool, about 1 hour. (Covered in plastic, the cakes can be left at room temperature for a few hours.) Prepare the frosting.

In a 3 or 4-quart nonreactive pan whisk together the sugar, flour, cornstarch and salt. Slowly whisk the in the coconut milk until smooth. Cook over medium heat, whisking constantly and scraping the corners of the saucepan, until the mixture boils and is very thick, about 6-10 minutes. Transfer the mixture to a wide bowl or glass pie plate and let cool completely, about 2 hours.

Using a stand mixer fitted with a whisk attachment, beat the butter (which should be between 65-68 degrees F) on medium-high speed until light and fluffy, about 5-minutes. Turn off the mixer, scrape the sides and bottom of the bowl and add the cooled coconut milk-flour mixture. Mix on medium speed until combined. Increase speed to medium-high and beat until the frosting is light and fluffy, 3 to 5 minutes.

Reserve 1 cup of the coconut frosting in a smaller bowl for decorations. Place one cake layer on a cake board or plate. Use 1-1/2 cups of frosting for the center and evenly spread across the layer. Place the second cake layer on top. Crumb coat the cake by spreading a thin layer of frosting over the entire layer of the cake to lock in the crumbs. This technique will prevent cake crumbs from getting mixed into the final frosting. Frost the cake with the remaining frosting.

To coat the cake with the shredded coconut, place coconut in your hand. Hold the edge of your hand close to the bottom of the cake, tilt your hand up and lightly press the coconut into the frosting. Repeat to coat the sides of the cake .