Preheat oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan.
In a large bowl, cream together the butter and sugar until light and fluffy. Beat in eggs one at a time, then stir in the vanilla.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Gradually add to the wet ingredients, mixing until just combined.
Stir in the mashed bananas.
In a small bowl, beat together the cream cheese and powdered sugar until smooth. Spread half of the batter into the prepared pan. Top with the cream cheese mixture, then spread the remaining batter over the top.
Bake for 35-40 minutes, or until a wooden skewer inserted into the center comes out clean.
Let cool completely before serving.
Notes
This cake is best served at room temperature. Store leftovers in an airtight container at room temperature for up to 3 days.