Preheat oven to 350°. Stir together first 5 ingredients in a large bowl; stir in bananas and next 5 ingredients, stirring just until dry ingredients are moistened. Pour batter into 4 greased and floured 9-inch square or round cake pans.
Bake at 350° for 20 to 25 minutes or until a wooden pick inserted in center comes out clean. Cool in pans on wire racks 10 minutes; remove from pans to wire racks, and cool completely (about 1 hour).
Spread Cream Cheese Custard Filling between layers. Spread Browned Butter Frosting on top and sides of cake. Chill 1 hour before serving. Store in refrigerator.