How to Make It
Prepare the Cake: Preheat oven to 350°F. Generously coat a 10-inch (10- to 15-cup) Bundt pan with cooking spray; set aside. Stir together flour, baking powder, baking soda, and salt in a bowl; set aside.
Beat butter, sugar, and vanilla in a stand mixer fitted with a paddle attachment on medium speed until light and fluffy, about 4 minutes. Add eggs, 1 at a time, and egg yolk, beating well after each addition. Reduce mixer speed to low; gradually beat in rum. Add flour mixture to egg mixture in thirds, alternately with heavy cream, beginning and ending with flour mixture, beating on low after each addition.
Transfer batter to prepared Bundt pan, smoothing the top. Bake in preheated oven until golden brown and a wooden pick inserted in center comes out clean, 50 to 55 minutes (cover loosely with aluminum foil after 30 minutes to prevent excess browning, if needed). Remove from oven.
During final 10 minutes of bake time, prepare the Rum Syrup: Melt butter in a saucepan over medium. Add sugar and 1/4 cup water; let mixture come to a boil. Boil, stirring occasionally, until syrupy, about 5 minutes. Carefully add rum and vanilla. (Mixture will bubble.) Boil, stirring often, 1 minute. Remove from heat.