Preheat the oven to 350°F (175 ℃). Grease a 9×9 inch light metal baking pan and line it with parchment paper.
In a medium bowl, whisk together the flour, baking soda, and salt. Then set aside the flour mixture.
2 1/4 cups (282 g) all purpose flour, 1 1/4 tsp baking soda, 1/2 tsp salt
Whip the smaller portion (10 tbsp/140 g) of cooled brown butter with an electric mixer on high speed until pale and fluffy. (If using a stand mixer, use a paddle attachment.) 10 tbsp (140 g) brown butter from above
Add the sugar to the whipped brown butter. Cream together for 2 minutes on high speed.
1 1/2 cups (300 g) granulated white sugar
Then add in the eggs and vanilla and mix on medium speed until pale in color and smooth, about 1 minute.
3 eggs, at room temperature, 1 1/2 tsp vanilla
Add in the buttermilk and mashed bananas. Mix on medium-low speed until combined. (The batter will look split at this point, but don't worry. It will pull together once the dry ingredients are added in.)
1 cup (240 ml) buttermilk, at room temperature, 1 1/4 cups (288 g) mashed overripe bananas, about 3 large bananas/4 medium bananas
Add in the dry ingredients, mixing on low speed until all is combined and the batter is smooth. Scrape the sides of the bowl with a rubber spatula as necessary.
Pour the batter into the prepared baking pan. Bake the cake for 40-45 minutes, or until a toothpick comes out clean from the center.
Let the cake cool in the pan on a wire rack for 30 minutes, then pull up on the parchment paper to release it from the pan. Let the cake continue cooling on the wire rack until completely cooled.