Heat the oven to 350° F, and adjust an oven rack to the middle position. Butter and flour two 8-inch cake pans.
In a medium bowl, whisk together the flour, baking soda, baking powder, and salt.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium until creamy. Add the sugar, and then beat again until light and fluffy, 3-5 minutes. Add the egg yolks one at a time, and mix until incorporated, scraping down the bowl as needed. Stir in the bananas, vanilla, and rum, then stir in the buttermilk until well combined.
Add the flour mixture, along with the nuts, to the batter, and stir gently until just combined.
Put the egg whites in a clean, dry bowl of a stand mixer fitted with the whisk attachment. Whisk until soft peaks form. Gently fold the egg whites into the batter.
Divide the batter evenly between the pans, and bake for 25 to 30 minutes or until a toothpick inserted into the center of the cakes comes out with only a few dry crumbs.
Move the pans to a wire rack and let the cakes cool in the pans for 20 minutes. Remove the cakes from the pans and let them finish cooling on the wire rack.