Like a genoise, this cake gets its spongy texture from eggs that are whisked until thick and voluminous.
Make the cake: Preheat oven to 350 degrees. Butter a 9-by-13-inch glass or ceramic baking dish.
Heat butter in a small saucepan over medium heat, swirling occasionally, until butter browns, 6 to 8 minutes. Remove from heat, and stir in vanilla.
Whisk egg whites, baking soda, and salt with a mixer on medium-high speed until soft peaks form, 2 to 3 minutes. Add yolks, and whisk until combined. Reduce speed to medium. Add sugar in a slow, steady stream, and mix until combined. Using a rubber spatula, quickly and gently fold butter mixture into egg mixture.
Combine flour and cinnamon in a small bowl. Sift 1/3 cup flour-cinnamon mixture over egg mixture, and gently fold to combine. Working quickly, sift remaining flour-cinnamon mixture over egg mixture in 2 additions, and gently fold to combine. Pour batter into prepared dish, and bake until golden and a tester inserted into center comes out clean, 20 to 25 minutes.
Meanwhile, whisk together the 3 milks in a medium bowl. Pour over cake as soon as it comes out of the oven. Transfer cake in dish to a wire rack to cool completely. Cover, and refrigerate overnight.
Make the rum-fruit topping: Bring rum, sugar, and water to a boil in a small saucepan, and cook until sugar dissolves, 1 to 2 minutes. Let cool for 30 minutes. Toss syrup with fruit. Serve cake with rum-fruit mixture and whipped cream.
To keep the airy batter from deflating, work gently but quickly as you fold in the browned butter and dry ingredients.