Instructions

1. 1
   Preheat the oven to 350°F. Butter a 9 x 13-inch cake pan.
2. 2
   In the bowl of an electric mixer, beat the eggs and sugar until pale and fluffy.
3. 3
   Add the flour, baking powder, and vanilla extract; mix until just combined.
4. 4
   Transfer the batter to the prepared pan. Bake 30–35 minutes at 350°F or until a toothpick inserted in the center tests clean.
5. 5
   Cool cake completely on a wire rack. Meanwhile, make coconut cream filling.
6. 6
   In a large bowl, whisk together sugar, coconut milk, yolks, shredded coconut, and cornstarch.
7. 7
   In a large saucepan, combine milk and remaining 1 cup sugar; heat over medium until sugar dissolves and milk is steaming.
8. 8
   Gradually whisk the hot milk into egg mixture; return the custard mixture to the saucepan.
9. 9
   Cook, stirring, until custard thickens, 3 to 4 minutes.
10. 10
    Pour the custard into a heat-proof bowl. Cover the surface directly with plastic wrap. Set aside to cool completely, about 1 hour (or up to overnight in the refrigerator).
11. 11
    While coconut cream and cake cool, make cake syrup.
12. 12
    Combine water and sugar in a medium saucepan; bring to a boil.
13. 13
    Transfer to a bowl with coconut milk and rum; stir to combine.
14. 14
    Combine sugar and water in a small saucepan and bring to a boil. Boil, stirring to dissolve the sugar, for 4 minutes.
15. 15
    In the bowl of an electric mixer, beat the egg whites and remaining 1 tablespoon sugar.
16. 16
    With the motor running, gradually beat in the syrup mixture in a slow, steady stream.
17. 17
    Preheat the oven to 400°F.
18. 18
    Use a serrated knife to cut cake horizontally into 2 layers.
19. 19
    Brush the bottom layer with syrup and spread with coconut cream.
20. 20
    Add the second cake layer and repeat with syrup and coconut cream.
21. 21
    Spread the meringue over the cake.
22. 22
    Transfer the cake to the oven for 2 to 3 minutes or until the meringue is toasted.