In a large mixing bowl, beat ½ cup of softened butter and 1 cup of granulated sugar together using an electric mixer on medium speed until creamy and light, about 2-3 minutes. Add 2 large eggs, one at a time, followed by 1 teaspoon of vanilla extract, mixing until smooth.

In a separate bowl, whisk together 2 cups of all-purpose flour, 1 teaspoon of baking soda, and ½ teaspoon of salt. Gradually add to the wet ingredients, mixing on low speed until just combined.

Gently fold in 1½ cups of mashed ripe bananas until evenly distributed.

Pour the batter into a greased 13x9-inch baking pan, spreading it evenly. Bake in a preheated oven at 325°F (160°C) for 35-40 minutes.

Remove the cake from the oven and cool for about 10 minutes in the pan before transferring to a wire rack.

In a saucepan, melt ½ cup of cubed cold butter over low to medium heat until golden brown, approximately 5-7 minutes. Allow it to cool slightly, then mix with 2 cups of powdered sugar, ¼ cup of heavy cream, and 1 teaspoon of vanilla extract until smooth.

Once the cake is completely cool, spread the brown butter icing generously over the top. Allow to set before slicing and serving.