Preheat the oven to 350°F (175°C). Grease and flour an 8-inch (20 cm) round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, beat the softened unsalted butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Add the mashed bananas to the wet mixture and mix until combined.
Alternately add the dry ingredients and the milk to the banana mixture, beginning and ending with the dry ingredients. Mix on low speed until just combined after each addition.
Fold in the sour cream gently with a spatula until the batter is smooth and uniform.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 35 to 40 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack and let it cool completely before slicing and serving.