Preheat oven to 350 degrees; position a rack in the center of the oven.
Add the butter to a 9-inch-round baking pan. Place the pan inside of the pre-heated oven until melted, about 5 minutes.
Remove the pan from the oven and add the brown sugar and rum; stir until mixed well and even.
Arrange the pineapple slices evenly in the pan; place a cherry in the center of each one and fill the center and any other open space with either additional pineapple pieces or pecans.
In a medium bowl, combine flour, baking powder and salt and whisk to combine.
Using a stand mixer, beat the egg whites until fluffy; scrape into a clean bowl.
Using the same mixing bowl, beat the egg yolks and sugar until thick and creamy; add the rum and vanilla extract and mix well..
Add the flour mixture and beat until well combined.
Using a spatula, fold in the egg whites until just mixed in.
Pour the batter over the fruit in the cake pan and smooth to the edges.
Bake for 40 minutes until the cake is golden brown and a toothpick inserted into the center comes out clean.
Place on a wire rack to cool for 10 minutes.
Run a knife around the edges of the pan to loosen the cake. Place a serving plate over the pan and invert carefully allowing the cake to slide out of the pan.
Serve warm or at room temperature.
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