Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake 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 and granulated sugar together with an electric mixer until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Mix in 1 teaspoon vanilla extract.
In a separate bowl, combine the mashed bananas, sour cream, and milk.
Gradually add the dry ingredients to the butter mixture alternately with the banana mixture, beginning and ending with the dry ingredients. Mix until just combined.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes. Then transfer to a wire rack to cool completely.
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.
Add 1 teaspoon vanilla extract and gradually beat in the powdered sugar until the frosting is smooth and spreadable.
Once the cake is completely cool, spread the cream cheese frosting evenly over the top of the cake.
Slice and serve.