Preheat oven to 375°F (190°C). Spray an 8-inch square pan with cooking spray.
Whisk together the flour, baking powder, and salt. Set aside.
Using an electric mixer, beat the butter, oil, and sugar. Beat in the eggs and egg white, 1 at a time, mixing well after each addition. Mix in the vanilla.
In a separate bowl, mash the bananas with the milk. Set aside.
Gradually mix half of the dry ingredients into the butter mixture. Mix in the bananas. Mix in the remaining dry ingredients. Pour the cake batter into the prepared baking pan.
Bake 40-45 minutes, or until a toothpick inserted into the center of the cake comes out clean. Cool the cake in the pan on a wire rack.
To make the frosting:
Using an electric mixer on medium speed, beat the cream cheese and vanilla. Reduce the speed and gradually add the confectioners’ sugar.
Spread the frosting on the cake. Sprinkle with nuts, if using.