Preheat an oven to 350 degrees F. Place an oven rack in the center position. Lightly but thoroughly grease a 12-cup bundt pan (I recommend using shortening or butter), then dust the inside with flour and tap out the excess. Set aside.
In a medium-sized bowl, whisk together the flour, brown sugar, cornstarch, kosher salt, and baking soda. In a separate medium-sized bowl, whisk together the sour cream, heavy cream, dark rum, and vanilla.
In a stand mixer fitted with the paddle attachment, cream the butter and granulated sugar together on medium-low speed for about 3-4 minutes, until the mixture starts looking less dense and more fluffy (this is a high sugar to butter ratio). Turn the mixer to high speed and cream for an additional 2-3 minutes. Scrape down the sides.
With the mixer on low speed, add the eggs and yolks, one at a time, allowing each to incorporate before adding the next but without over-mixing. As soon as the egg has blended into the batter, move onto the next one. Stop and scrape down the sides of the bowl a couple times during the process.
With the mixer on medium-low speed, swiftly alternate between adding the dry and wet ingredients, starting and ending with the dry. This shouldn’t take more than 60 seconds total. Once the ingredients have been added, turn off the mixer and give a few good stirs with a spatula to make sure the ingredients are combined.
Pour into the prepared bundt pan and level off the top with an offset or regular spatula. Bake for 1 hour 10 minutes, or until a skewer comes clean from the center.
While the cake is baking, prepare the rum syrup. In a medium-sized saucepan, combine the butter, water, sugar, salt, 3/4 cup rum, and vanilla. Bring to a boil over medium-high heat while stirring periodically, then turn the heat to low and simmer for 6-8 minutes, until the syrup has thickened slightly. Remove from the heat and stir in 1 or 2 remaining tablespoons of rum to taste (depending on how aggressive you want the rum flavor in the cake to be). Allow to cool in the pan for 10 minutes, then invert to a wire rack to cool almost completely, around 45 minutes. While the cake is cooling, clean and dry the bundt pan.
Once the cake has cooled, fit the clean bundt pan on top of it and then gently turn the cake back over and remove the wire rack. Briefly whisk the syrup incase any separation has occurred. Use a skewer to poke several holes in the cake, then slowly and evenly top with the rum syrup, pausing as needed so it can absorb into the cake. Let the syrup soak for 1 hour, then turn the cake out onto a stand for serving.