To make the cake: Preheat the oven to 350 degrees F. Butter a 9 by 13-inch cake pan. In the bowl of an electric mixer, beat the eggs and sugar until pale and fluffy. Add the flour, baking powder, and vanilla extract; mix until just combined. Transfer the batter to the prepared pan. Bake 30 to 35 minutes or until a toothpick inserted in the center tests clean. Cool completely on a wire rack. Meanwhile, make coconut cream filling.
For the coconut cream filling: In a large bowl, whisk together 3/4 cup sugar, coconut milk, yolks, shredded coconut and cornstarch. In a large saucepan, combine milk and remaining 3/4 cup sugar; heat over medium until sugar dissolves and milk is steaming. Gradually whisk the hot milk into egg mixture; return the custard mixture to the saucepan. Cook, stirring, until custard thickens, 3 to 4 minutes. Pour the custard into a heat-proof bowl. Cover the surface directly with the plastic wrap. Set aside to cool completely, about 1 hour (or up to overnight in the refrigerator). While coconut cream and cake cool, make cake syrup.
For the simple syrup: Combine water and sugar in a medium saucepan; bring to a boil. Transfer to a bowl with coconut milk and rum; stir to combine.
For the meringue: Combine 3/4 cup sugar and water in a small saucepan and bring to a boil. Boil, stirring to dissolve the sugar, for 4 minutes. In the bowl of an electric mixer, beat the egg whites and remaining 1 tablespoon sugar. With the motor running, gradually beat in the syrup mixture in a slow, steady stream.
To assemble the cake: Preheat the oven to 400 degrees F. Use a serrated knife to cut cake crosswise into 2 layers. Brush the bottom layer with syrup and spread with coconut cream. Add the second cake layer and repeat with syrup and coconut cream. Spread the meringue over the cake. Transfer the cake to the oven for 2 to 3 minutes or until the meringue is toasted.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.