Pre-heat and prepare the pans: Preheat oven to 350- making sure rack is placed in center of oven. Butter the bottom and sides of two 9 inch round cake pans. Line the bottoms with parchment paper and butter the paper. Dust the sides with flour, tapping out any excess flour.
Prepare the cake: Cream the butter and 1 cup of the sugar in a large bowl with an electric mixer on medium speed until it is light and fluffy, 3-4 minutes.
Add the coconut extract then the egg yolks, one at a time- beating well after each addition. Scrape the bowl often.
Sift the cake flour, baking powder, and salt together into another bowl. Add one third of the flour mixture to the butter mixture, beating on low speed until ti is incorporated. Add half of the coconut milk and beat until incorporated, scraping down the sides of the bowl.
Repeat with another third of the flour, followed by the remaining coconut milk. Add the remaining flour, scrape down the sides of the bowl and beat on medium speed until well incorporated (about 2 minutes.).
In another bowl beat the egg whites on low speed until they are foamy- abut one minute.
Add the remaining 1 cup sugar, increase the speed to high and whip the whites until soft peaks form- 3 to 4 minutes more.
Fold half of the whites gently into the cake batter, using a large rubber spatula. Combine the remaining whites by gently folding them into the batter.
Scrape the batter evenly into the prepared pans.
Bake until a wooden toothpick inserted in the center of the layers comes out clean, 20-30 minutes.
Let the layers cool in the pans on a wire rack for 10 minutes and them remove them from the pans and place on the rack to cool completely- at least 2 hours.
Carefully peel off the paper then split each layer in half horizontally with a serrated knife.
Wrap the layers with plastic wrap and chill in the refrigerator for at least two hours.
Buttercream: Place the egg yolks and butter in a bowl and beat with an electric mixer on high speed until the mixture is pale yellow, about 2 minutes.
Add the confectioners' sugar and salt and beat on low until well combined.
Increase the speed to high and beat until the mixture is creamy, about 2 minutes more.
Add the coconut milk and beat again until well combined and creamy, about 1 minute. Cover and refrigerate while you prepare the frosting.
Frosting: Place the sugar, egg whites, 1/4 cup water and cream of tartar in the top of a double broiler over simmering water. Beat with an electric mixer until very firm, 7 minutes.
Remove from heat, add the vanilla and beat until the frosting is cooled and holds a stiff peak.
Assemble the cake: Place 1 cake layer (cut sides down) on a serving plate.
Spread it with 1/3 of the buttercream.
Repeat with 2 more layers and the remaining buttercream.
Top with the remaining layer, smooth side up.
Immediately frost sides and top of cake with frosting. A frosting knife works best to make attractive swirls.
Keep at room temperature in a cake server until serving.
****IMPORTANT NOTE: This buttercream uses raw egg yolks which can cause heath problems - Be sure to use VERY fresh eggs which have been kept refrigerated!