Preheat the oven to 350 degrees F. In a small baking dish, lined with foil (for easy clean up), place the banana(s) and roast for 20 minutes. You should see juices pooling on the bottom of the dish. Carefully peel the bananas and measure out ½ a cup (roughly 4 ounces) of roasted banana. Mash in a small bowl, set aside, and let cool to room temperature.
Keep the oven at 350 degrees F (as it was for roasting the banana). And grease and line an 8x8-inch baking pan. (See below in the “NOTES” section for other baking pan alternatives).
In a medium bowl, sift together the flour, baking soda, and salt. Set aside.
In the bowl of a stand-up mixer with the whisk attachment (alternatively you can do this in a large bowl with an electric mixer), add the egg whites and beat on medium for 2 minutes or until it is foamy. Pour in the sugar, increase the speed to medium-high, and beat for another 2 minutes or until it becomes opaque white and starts to look slightly airy. You’re not making meringue, so no need to look for “peaks.”
Replace the whisk attachment with a paddle if using a stand mixer and turn the mixer to low. Add in the butter, oil, buttermilk, vanilla, and mashed banana and mix until incorporated. Add the dry mixture and beat until combined and smooth.
Pour the batter into the prepared baking pan and smooth it out until it reaches the sides. Transfer to the oven to bake for 25 to 32 minutes, until a skewer inserted into the center comes out clean.
Allow the cake to cool in the pan for about 10 minutes and then invert onto a cooling rack and cool to room temperature. It’ll probably take a good hour.