Preheat the oven to 350°F (175°C). Grease and flour a 9×13 inch sheet pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and light brown sugar together until light and fluffy, about 3-4 minutes.
Add the eggs one at a time, beating well after each addition. Stir in 1 teaspoon vanilla extract.
Mix in the mashed ripe bananas until combined.
Alternately add the dry flour mixture and the buttermilk to the banana mixture, beginning and ending with the flour mixture. Mix just until combined after each addition.
Fold in the sour cream until the batter is smooth.
Pour the batter evenly into the prepared sheet pan and spread it out smoothly.
Bake for 28-32 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool completely in the pan on a wire rack.
While the cake cools, prepare the cream cheese frosting: In a medium bowl, beat the softened cream cheese and 1/4 cup butter together until smooth and creamy.
Gradually add the powdered sugar, beating until smooth and fluffy.
Add 1 teaspoon vanilla extract and 2 tablespoons milk, and beat until the frosting reaches a spreadable consistency.
Once the cake is completely cooled, spread the cream cheese frosting evenly over the top.
Slice into 12 squares and serve.