Preheat Oven: Set your oven to 350°F (180°C) and position the rack in the center.

Combine Wet Ingredients: In a liquid measure, whisk together the milk, sour cream, and rum. Let it come to room temperature.

Cream Butter and Sugar: In a large bowl or stand mixer, whip the butter on medium speed for 1 minute. Add granulated sugar and beat for 3–4 minutes until light and fluffy. Mix in vanilla extract. Add eggs and egg yolks one at a time, mixing briefly after each addition. Scrape the sides of the bowl.

Mix Dry Ingredients: In a separate bowl, combine cake flour, baking powder, baking soda, and salt.

Alternate Ingredients: Add 1/3 of the dry ingredients to the butter mixture, mixing until just combined. Add half of the milk/sour cream/rum mixture and mix lightly. Repeat, alternating remaining dry and wet ingredients, finishing with the last 1/3 of the dry ingredients. Scrape down the sides to ensure even mixing.

Bake Cake: Grease and flour a bundt pan thoroughly. Pour batter evenly into the pan. Bake for about 50 minutes, or until a toothpick inserted in the center comes out clean or with a few crumbs.

Prepare Glaze: In a saucepan over medium heat, combine butter, granulated sugar, brown sugar, rum, heavy cream, and vanilla. Stir until smooth, bring to a boil, and boil for 2 minutes while stirring.

Finish Cake: Let the cake cool in the pan for 15 minutes. Gently invert onto a plate, then return it to the pan. Poke holes all over the cake and pour most of the glaze over it, reserving a little for topping. Allow it to soak in.

Serve: Spoon remaining glaze on top before serving. Store leftovers covered in the refrigerator.