Preheat the oven to 350°F (175°C). Grease a 9 x 13-inch baking pan and line with parchment paper if desired
Whisk together the flour, baking soda, baking powder, and salt in a medium bowl
In a large bowl, beat the softened butter, granulated sugar, and brown sugar until light and fluffy
Add the eggs one at a time, beating well after each addition. Mix in the vanilla
Stir in the mashed bananas until evenly combined
Add the dry ingredients in batches, alternating with the buttermilk. Mix just until the batter comes together
Spread the batter evenly in the prepared pan
Bake for 30 to 35 minutes, or until the top is lightly golden and a toothpick inserted in the center comes out clean or with a few moist crumbs
Cool the cake completely on a cooling rack
For the frosting, melt the butter in a saucepan over medium heat. Continue cooking until golden-brown bits form and the butter smells nutty and toasted
Immediately transfer the browned butter to a heatproof bowl and let it cool until warm, not hot
Beat the browned butter with powdered sugar, cream or milk, rum, vanilla, and salt until smooth and creamy. Adjust the consistency with a little more powdered sugar or liquid as needed
Spread the frosting over the completely cooled cake
Slice and serve at room temperature