Preheat oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan.

In a medium bowl, whisk together flour, baking soda, baking powder, and salt.

In a large bowl, cream together butter and sugars until light and fluffy. Beat in eggs one at a time, then stir in mashed banana and vanilla.

Gradually add dry ingredients to wet ingredients, alternating with milk and beginning and ending with dry ingredients. Mix until just combined.

Pour batter into prepared pan and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Let cool completely before frosting.

In a small saucepan, melt ½ cup of butter over medium heat. Cook until golden brown and nutty aroma develops, then remove from heat and let cool slightly.

In a large bowl, beat together browned butter and cream cheese until smooth. Gradually add powdered sugar, vanilla, and salt, beating until light and fluffy.

Frost cooled cake with the brown butter cream cheese frosting.