Make the cake: Preheat the oven to 350 degrees F. Butter and flour two 9-inch-by-2-inch round cake pans and line the bottoms with parchment paper. Whisk together the milk, egg whites, vanilla seeds and vanilla extract in a medium bowl.

In the bowl of a stand mixer fitted with the paddle attachment, mix together the 2 1/4 cups flour, the sugar, baking powder and salt. With the mixer running on low speed, add the 12 tablespoons butter, one piece at a time, and continue beating until the mixture resembles moist crumbs. Add all but 1/2 cup of the milk mixture and beat on medium speed until pale and fluffy, about 1 1/2 minutes. With the mixer on low speed, add the remaining 1/2 cup milk mixture. Increase the speed to medium and beat for 30 seconds more. Scrape the sides of the bowl and mix for 20 seconds longer. Divide the batter evenly between the cake pans and smooth the tops with a rubber spatula.

Bake until a toothpick inserted into the center comes out with a few crumbs attached, 22 to 24 minutes. Cool in the pans on a baking rack for 10 minutes. Run a small knife around the sides of the pans and invert the cakes onto the baking rack. Remove the parchment paper and let cool completely, about 45 minutes.

Make the coconut filling: Combine the coconut custard and cream in a bowl and whip until soft peaks form.

Make the coconut buttercream: Beat the butter and confectioners' sugar in a stand mixer fitted with the paddle attachment until light and fluffy, about 4 minutes. Add the coconut custard and salt; beat until the buttercream is smooth.

Assemble the cake: Use a long serrated knife to slice each cake horizontally into 2 layers. Reserve one of the flat bottom layers for the top of the cake. Place another one of the layers on a cardboard round, cut-side up, and brush with some of the coconut simple syrup. Spoon one-third of the coconut filling onto the cake, and using a small offset metal spatula, spread it into an even layer, leaving a 1/2-inch border around the edge of the cake. Repeat with 2 more layers. Brush the cut side of the reserved cake layer with the remaining syrup. Place the layer, cut-side down, on top of the cake. Frost the sides and the top of the cake with the buttercream. Pat the toasted coconut onto the sides of the cake and sprinkle some on the top.

Photograph by Yunhee Kim