Butter three 9-inch by 2-inch round cake pans. Line with round parchment paper; butter parchment, dust lightly with flour and shake off excess.
Sift together cake flour, all-purpose flour, baking powder and salt.
In a separate bowl, combine unsweetened coconut milk and vanilla; whisk to blend, add coconut and stir to blend.
In a bowl of a stand mixer fitted with a paddle attachment, beat butter and 1½ cups sugar for 3 minutes, or until light and fluffy. Add eggs, one at a time, and mix until incorporated; transfer batter to a large bowl.
Alternately add flour mixture and coconut mixture to batter in 3 batches, until blended.
In the bowl of a stand mixer fitted with a wire whip, beat egg whites with cream of tartar and 1½ cups sugar to form medium stiff peaks; add to batter, stirring slowly with a balloon whisk until incorporated.
Spread batter evenly in pans; tap pans on counter to eliminate air bubbles.
Bake for 35 minutes, or until a toothpick inserted in center comes out clean; cool on cooling racks for 5 minutes. Run a knife around outer edges of pans and invert cakes onto racks
Coconut Simple Syrup
Combine unsweetened coconut milk, cream of coconut, coconut rum and simple syrup and whisk to blend. Reserve.
Coconut Cake Filling
In bowl of stand mixer filled with a wire whip, combine heavy cream, cream of coconut, confectioner’s sugar, shredded coconut and vanilla extract.
Whip until stiff peaks form.
Arrange 1 cake layer on a cake stand or large plate. Brush the top of cake generously with reserved coconut simple syrup. Using an offset spatula, spread evenly with coconut cake filling. Repeat with 2nd and 3rd cake layer.
Chill cake for 3 hours before serving.
Serving the Coconut Cake
This cake is best served very cold, so be sure to wait until the last minute to pull it out of the refrigerator.
To add a bit of color and extra flavor to the dessert course, add a handful of fruit on the side of each slice. Look for berries that are in season. Blueberries, strawberries and blackberries would all be delicious.