Make cake: Preheat oven to 375°F. Mist an 8-inch square pan with cooking spray. Mix flour, baking powder and salt. With an electric mixer, beat butter, oil and sugar. Beat in eggs and egg white, then vanilla. Mash bananas with milk.
Stir half of dry ingredients into butter mixture, then stir in banana mixture. Mix in remaining dry ingredients. Pour batter into baking pan. Bake until a toothpick inserted in center of cake comes out clean, about 40 to 45 minutes. Let cake cool in pan on a wire rack.
Make frosting: With an electric mixer, beat cream cheese and vanilla. Reduce speed and gradually beat in confectioners' sugar. Spread cake with frosting and sprinkle with walnuts, if desired.