Preheat oven to 350℉. Grease a 12-cup bundt cake pan and set aside.
Whisk together the dry ingredients–flour, baking powder, baking soda and salt. Set aside.
In a separate bowl or measuring cup, whisk together rum, buttermilk and vanilla. Set aside.
In the bowl of a stand mixer (or using a hand mixer), beat butter, sugar and brown sugar until creamy, about 2 minutes. Add the eggs and egg yolks, one at a time, mixing well after each addition. Scrape down the sides and bottom of the bowl as necessary.
Add one-third of the dry ingredients and mix just until combined. Add half the rum-buttermilk mixture and mix just until combined. Repeat, then end with remaining flour mixture. Scrape bottom and sides of the bowl as needed.
Transfer batter to prepared bundt cake pan. Bake at 350℉ for 45 to 55 minutes or until a toothpick inserted in the cake comes out with little to no crumbs.
To make the glaze, melt the butter with the heavy cream in a saucepan over medium-high heat. Once melted, add the sugar and bring mixture to a boil. Boil for about 1 minute, then reduce heat to medium-low and cook until thickened and sugar is dissolved, about 5 more minutes. Remove from heat and gradually stir in the rum.
Cool the cake in the pan for 10 minutes, then invert it on a wire rack just to make sure it comes out of the pan without a problem. Place the cake back in the pan and skewer the top, poking holes 1 inch apart all over the top.
While the cake is still warm and in the pan, pour half the glaze mixture over the top. If it pools at all, poke a few more holes to help the sauce soak in. Let it sit for 5 minutes, then invert the cake onto a wire rack or serving plate. Use a pastry brush to brush the remainder of the sauce over the outside of the cake. Let it soak in before applying more to prevent it from dripping down the sides.
Let the cake cool completely before slicing and serving. Typically it tastes best the day after it's baked.
Notes
Dark rum is best. I suggest Kraken, Myers or Gosling's.Buttermilk substitute: Place 1 1/2 teaspoons white distilled vinegar in a liquid measuring cup. Add milk and fill to the 1/2 cup mark. Stir and let the mixture sit 10 minutes before using.Store covered at room temperature for up to 1 week or in the refrigerator for up to 2 weeks.Freeze for up to 3 months. Once the cake has cooled, wrap the whole thing (or slices) in plastic wrap and aluminum foil to prevent freezer burn. Freeze and, when ready to eat, defrost in the refrigerator.