Preheat oven to 350° F with the rack in the center.
Get your eggs out of the refrigerator first and let them come up to room temperature.
Spray two 9″ cake pans lightly with cooking oil spray. Use round pieces of parchment paper, or cut round pieces, and place them in the bottoms of the pans. Hit the paper lightly with a little more spray.
Put the coconut milk in a small bowl and measure in the shredded or flaked coconut. Stir in the coconut until smooth.
Separate all of the eggs, putting the whites into the bowl of your stand mixer and holding the yolks aside for now in a bowl or cup.
Use the whisk attachment in your stand mixer on high speed until the egg whites are foamy with a consistency like whipped cream. Put the whipped whites into a large bowl and set aside.
Now put the yolks into the bowl of your stand mixer. Add the sugar and salt.
Melt the butter in the microwave or a sauce pan. Add the butter to the mixture. Beat with the paddle attachment on speed 4 for 1 minute. Stop beating when you see a smooth blended consistency.
Weigh or measure the flour in a bowl. Add the baking powder to the flour and stir completely into the flour with a spoon.
Add the flour mixture to the mixer bowl. Add the coconut mixture. Beat with the paddle attachment on speed 4 for two minutes until smooth and combined. Use your flexible spatula to clear the paddle and scrape down the sides of the bowl, being sure to push all the way to the bottom.
Remove the bowl from the stand mixer and add the egg whites. Using your spatula, fold the batter carefully until all of the egg white mixture is mixed together with the batter. You don’t want to blend them. Be careful not to deflate the egg whites as you fold them in.
Dump the batter into the two prepared cake pans. Use your kitchen scale to weigh them, and spoon batter from one to the other until they are the same weight. Place the cake pans into the preheated oven just back of the center. Bake 15 minutes.
After 15 minutes, switch the two pans and turn them 180° in the oven so they get done evenly. Bake an additional 15 to 25 minutes.
Check the cakes by sticking a toothpick deeply into the center. Add time in increments of at least 2 minutes so the oven doesn’t cool down from opening the door.
Take the cakes out of the oven and allow to cool 5 minutes.
Use the dull side of a table knife to be sure the cakes aren’t stuck to the side of the cake pans. Invert a wire rack over each cake. Flip the cake upside down, rack, pan, and all. If the cake doesn’t drop down onto the rack, thump the middle of the cake pan with the heel of your hand. Remove the pans and the parchment paper. Allow the layers to cool completely on the wire racks. Do not proceed with frosting until the cakes are completely cool to the touch.
Make Buttercream Cheese Frosting.
Put one of the layers flat side down on a plate or cake circle. Put four small pieces of wax paper under the layer from four sides so the frosting doesn’t get beyond the cake.
Scoop enough frosting to separate the layers by 3/8 inch onto the first layer. Spread the frosting out with a spatula or table knife.
As a little surprise layer, optionally spread 1/4 to 1/2 cup fruit jam or preserves on the bottom of the second layer.
Center the second layer flat side down over the first layer, and push down until the frosting reaches the edge. Scoop the rest of the frosting on top of the second layer and spread it out with a spatula. Use a table knife to bring gobs of frosting from the top to the sides of the cake and spread it out. Turn the cake against the knife to smooth the sides. Spread the frosting on top until it is even.
Place about 1/2 cup shredded or flaked coconut in a skillet on medium high heat. Move it around and turn it with your spatula so it won’t burn.
When it gets to the point where you see some browning, start to move and turn it continuously.
When it gets to the point where about half the coconut is lightly browned, pick up the pan and get it away from the heat. Immediately and quickly start scraping the coconut into a bowl. It will still be browning from the hot pan.
To get the coconut to stick to the sides of the cake, you basically have to pelt and pummel it repeatedly with thrown coconut. Sprinkle the remainder on top. Remove the wax paper.