Preheat oven to 325 degrees F.

Spread the coconut evenly onto a baking sheet and toast until lightly golden brown, stirring once, 8 to 10 minutes. Turn off the oven and let the coconut sit in the oven until very dry and crunchy, about 15 minutes longer.

For the simple syrup:

Bring water and sugar to a boil. Stir in the coconut, remove from the heat and let sit for at least 30 minutes and up to 4 hours. Strain the liquid into a clean saucepan, bring to a boil and let cook until the mixture is slightly reduced, about 5 minutes. Let cool.

For the custard:

Combine the milks and vanilla bean and seeds in a medium nonreactive saucepan and bring to a simmer over low heat.

Whisk together the yolks, sugar and cornstarch in a large bowl. Slowly whisk the warm milk into the egg mixture then return the mixture to the pot over medium heat and bring to a boil, whisking constantly, until thickened. Scrape the mixture into a bowl and whisk in the rum and vanilla extract. Let cool to room temperature then cover with plastic wrap and refrigerate until cold, at least 2 hours.

For the filling:

Combine the custard and cream in a bowl and whip until soft peaks form.

For the buttercream:

Beat the butter and sugar in a stand mixer fitted with the paddle attachment until light and fluffy, about 4 minutes. Add the coconut custard and salt and beat until combined and smooth.

For the cake:

Preheat oven to 350 degrees F. Butter and flour 2 (9 by 2-inch) round cake pans and line bottoms with parchment paper.

Whisk together the milk, egg whites, vanilla bean seeds and vanilla extract in a medium bowl.

In the bowl of a mixer fitted with the paddle attachment, mix together the flour, sugar, baking powder and salt. With mixer running at low speed, add the butter, one piece at a time and continue beating until mixture resembles moist crumbs. Add all but 1/2 cup of milk mixture to crumbs and beat at medium speed until the mixture is pale and fluffy, about 1 1/2 minutes. With mixer on low speed, add remaining 1/2 cup of the milk mixture, increase speed to medium and beat 30 seconds more. Scrape sides of bowl and mix for 20 seconds longer. Divide the batter evenly between the cakes pan and smooth the tops using 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 pan on baking rack for 10 minutes. Run a small knife around the side of the pan and invert cakes onto the baking rack, removing parchment paper, and let cool completely, about 45 minutes.

To Assemble:

Using a long serrated knife, slice each cake horizontally into 2 layers. Reserve 1 of the flat bottom layers for the top of the cake. Place another layer on a cardboard round cut side up and brush with some of the coconut simple syrup. Spoon 1/3 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 top of the cake with the buttercream. Pat the coconut onto the sides of the cake and sprinkle the remaining coconut on the top of the cake.