Preheat oven to 400°.
To prepare bars, combine banana, brown sugar, and 1 tablespoon butter in an 8-inch square baking dish. Bake at 400° for 35 minutes, stirring after 17 minutes. Cool slightly.
Reduce the oven temperature to 375°.
Weigh or lightly spoon cake flour into dry measuring cups; level with a knife. Combine 9 ounces (about 2 1/4 cups) flour, soda, and baking powder in a medium bowl. Combine banana mixture, buttermilk, and 1 teaspoon vanilla in another medium bowl. Place 1/2 cup butter and granulated sugar in a large bowl; beat with a mixer at medium speed until well blended. Add eggs to granulated sugar mixture; mix well. Add flour mixture to sugar mixture alternating with banana mixture, beginning and ending with flour mixture.
Pour batter into a 13 x 9–inch baking pan coated with baking spray. Bake at 375° for 20 minutes or until a wooden pick inserted in center comes out clean. Cool completely in pan on a wire rack.
To prepare frosting, melt 1/4 cup butter in a small saucepan over medium heat; cook 4 minutes or until lightly browned. Cool slightly. Combine browned butter, powdered sugar, cream cheese, and 1 teaspoon vanilla in a medium bowl; beat with a mixer until smooth. Spread frosting over cooled bars. Sprinkle with pecans.