Preheat oven to 350 degrees. Grease and flour a 10-inch Bundt pan.
Stir together flour, baking powder, baking soda, and salt in a medium bowl. Set aside.
Beat butter, sugar, and vanilla in a stand mixer fitted with a paddle attachment on medium speed until light and fluffy (3-4 minutes). Add eggs, 1 at a time, and egg yolk, beating well after each addition. Reduce mixer speed to low. Gradually blend 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.
Pour batter into bundt pan, smoothing the top. Bake 50-55 minutes or until a toothpick inserted in middle comes out clean.
Using a wooden skewer, poke warm cake all over in pan. Spoon ¾ cup of the rum sauce evenly over cake. Let stand 10 minutes. Invert cake onto a cake stand or plate.
Spoon remaining rum sauce evenly over the top of cake. Let cool at least 20 minutes before serving.
During final 10 minutes of bake time prepare the Rum Sauce:
Melt butter in a medium saucepan over medium heat.
Add sugar and ¼ cup filtered water. Bring mixture to a boil, stirring occasionally, until mixture becomes syrupy (about 5 minutes).
Remove pan from heat and carefully add rum and vanilla (mixture will bubble up). Blend well and return to heat. Return mixture to a boil, stirring constantly for 1 minute. Remove from heat.