Creating Brown Butter Banana Cake is a rewarding process. Follow these steps to ensure success:

1. Preheat the Oven: Preheat your oven to 350°F (175°C). Grease and flour two 9-inch round cake pans.
2. Brown the Butter: In a saucepan over medium heat, melt the butter. Continue to cook, swirling occasionally, until it turns golden brown and has a nutty aroma. Remove from heat and let cool slightly.
3. Mix Wet Ingredients: In a large bowl, combine the mashed bananas, granulated sugar, eggs, and vanilla extract. Stir until well mixed.
4. Add Brown Butter: Pour the browned butter into the banana mixture and stir until combined.
5. Combine Dry Ingredients: In another bowl, whisk together the flour, baking powder, baking soda, and salt.
6. Combine Mixtures: Gradually add the dry ingredients to the wet ingredients, alternating with buttermilk. Mix until just combined; do not over-mix.
7. Bake: Divide the batter evenly between the prepared cake pans. Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
8. Cool: Remove from the oven and let the cakes cool in the pans for 10 minutes. Then, transfer them to wire racks to cool completely.
9. Prepare Frosting: In a mixing bowl, beat the softened cream cheese and butter together until creamy. Gradually add the powdered sugar and vanilla, mixing until smooth. Add milk if needed for spreadability.
10. Assemble the Cake: Once the cakes are completely cool, place one layer on a serving plate. Spread a layer of cream cheese frosting on top. Place the second layer on top and frost the top and sides of the cake with the remaining frosting.