Preheat oven to 350°. Spray a 10-cup Bundt pan with baking spray with flour.

In a medium bowl, whisk together flour, baking powder, and salt; set aside. In another medium bowl, beat egg yolks and . cup (138 grams) sugar with a mixer at medium-high speed until thick and pale, about 2 minutes. Reduce mixer speed to low. Beat in vanilla bean seeds and rum extract.

Gradually add flour mixture to egg yolk mixture, beating just until combined after each addition. Add butter and cream, beating to combine.

In a large bowl, beat egg whites with a mixer at medium-high speed until frothy. Increase mixer speed to high, and gradually add remaining 1⁄2 cup (93 grams) sugar, a few tablespoons at a time. Beat until stiff glossy peaks form, about 4 minutes.

Add one-third of egg whites to yolk mixture, combining gently. Add remaining egg whites, folding gently to combine. Spoon batter into prepared pan.

Bake until a wooden pick inserted near center comes out clean, 45 to 50 minutes. Let cool in pan for 10 minutes. Do not remove from pan. Using a skewer, pierce cake 12 to 15 times. Pour 2 cups Rum Syrup onto cake. Let stand for 15 to 20 minutes.

Meanwhile, in a small saucepan, cook remaining 1 cup Rum Syrup over medium heat, stirring occasionally, until thickened, about 15 minutes. Remove from heat, and let cool completely. Invert cake onto serving platter. Drizzle with desired amount of thickened Rum Syrup, reserving remainder for another use, if desired. Store, covered, at room temperature for up to 5 days.