Preheat the oven to 350 degrees F. Coat the bottom and sides of an 8 X 8 inch square cake pan with cooking spray or softened butter. Cut a piece of parchment paper measured to fit the bottom of the cake pan, but long enough to extend a couple of inches above the pan on two opposite sides. Place into the pan, allowing the extra paper to extend above the two sides.
In a large bowl, whisk the melted butter, brown sugar, and vanilla extract together until thoroughly mixed. Add the beaten eggs and whisk to incorporate them. Add the buttermilk and the mashed bananas and mix well. Sprinkle the baking soda over the mixture and whisk to mix it in. Sprinkle the salt in and whisk well. Fold the flour in gently, using a flexible spatula, mixing only until you don’t see any evidence of flour in the batter. Do not overmix the batter.
Transfer the batter to the prepared cake pan. Smooth the top of the batter and make sure it is evenly into all four corners. Bake from 25 to 30 minutes, rotating the pan halfway through the baking time. Test the cake with a wooden skewer or toothpick for doneness. Only a crumb or two should adhere to the wooden tester.
Remove the cake from the oven and allow it to sit in the pan on a rack for 20 minutes. Use the parchment extensions to lift and transfer the cake to a rack and cool completely to room temperature. (If the cake sticks when you try to remove it, run a table knife around the perimeter of the pan to loosen it.) Make sure the cake is completely cooled before frosting it.