Make the cake:
Position a rack in the center of the oven and heat the oven to 350°F. Butter and flour two 9x2-inch round cake pans and line the bottoms with parchment.
In a medium bowl, whisk the flour, baking powder, and salt. In another medium bowl, whisk the eggs, yolks, oil, and vanilla.
In the bowl of a stand mixer fitted with the paddle attachment, mix the butter on medium-high speed until smooth and light in color, about 2 minutes. Scrape the sides of the bowl with a rubber spatula. Add the sugar and mix on medium-high speed until very well blended, about 2 minutes longer. Scrape the bowl again.
With the mixer running on medium-high speed, add the egg mixture to the butter mixture in a steady stream. Beat until light and fluffy, about 2 minutes.
Reduce the mixer speed to low. Add half the dry ingredients to the butter mixture and mix until combined. Add half the buttermilk and mix until combined. Repeat with remaining flour and buttermilk.
Scrape the sides of the bowl and divide the batter evenly between the prepared pans. The batter will be thick; spread it in the pans as smoothly as possible.
Bake the cakes until they're golden-brown on top and a toothpick inserted into the middle of each cake comes out clean, 30 to 35 minutes. Cool them in the pan for 15 to 20 minutes on a cooling rack and then turn them out onto a rack and cool completely.
Make the whipped ganache:
While the cake cools, prepare the ganache. Clean and dry the stand mixer bowl. Grind the chocolate in a food processor until it reaches the consistency of coarse meal, about 30 seconds. Bring the cream to a boil in a small saucepan over medium heat. Add the cream and rum to the food processor and process until smooth, 10 to 20 seconds. Transfer to the cleaned and dried mixer bowl and refrigerate, stirring occasionally, until the ganache reaches 55 to 65°F, about 1-1/2 hours.
In the stand mixer fitted with the whisk attachment, whip the chilled ganache on medium-high speed until lightened in color and fluffy, about 1 minute. Don't overwhip, or the ganache may seize. Scrape the sides of the bowl with a rubber spatula and mix gently.
Assemble the cake:
Put one cake layer on a cake plate and tuck strips of waxed paper under the cake to keep the plate clean. Using an offset spatula, spread about 2 cups of the whipped ganache over the top in an even layer right to the edges of the cake. Top with the second cake layer.
Brush any large, loose crumbs off the cake and quickly ice the top and sides with a thin layer of the whipped ganache to seal the cake. Spread most or all of the remaining ganache over the sides and top of the cake, using the spatula to decoratively dimple and swirl the icing. Carefully pull the wax paper from under the cake and discard. Store the cake in the refrigerator, but let it sit at room temperature for 20 to 30 minutes before serving.