Preheat oven to 350°F. Butter and lightly flour three 6-inch round cake pans. Line bottom with parchment paper. Set aside.
Whisk together cake flour, baking powder, and salt. Set aside.
In another bowl, mix together coconut milk, egg whites, and vanilla extract until smooth. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, beat together butter and sugar until smooth. Scrape down sides of bowl as needed to ensure thorough mixing.
Alternate adding dry and wet ingredients to the mixing bowl. Add ⅓ of dry flour mixture. Mix on on low speed while adding ½ of liquid mixture. Add another ⅓ dry mixture and follow with remaining ½ of liquid mixture. Add remaining ⅓ of dry mixture. Batter will be thick. Scrape down sides of bowl as needed.
Divide batter evenly among prepared cake pans ( about 390 grams of batter per pan). Use a mini offset spatula or the back of a spoon to spread batter into an even layer. Bake for 30-35 minutes until toothpick inserted in center of cake comes out clean. Cool cake in pan for 5 minutes. Run a mini offset or knife along cake pan edges to loosen cake. Unmold cake and cool to room temperature on wire rack.