Preheat the oven to 350℉ and grease or line a 9X9-inch baking pan with parchment paper.
In a large bowl using a handheld electric mixer, cream together the butter and sugar until combined. Add the eggs and vanilla and mix until smooth. Add the mashed bananas and milk and mix until all is combined.
½ cup salted butter, 1 cup brown sugar, 2 eggs, 1 teaspoon vanilla extract, 1 ½ cups mashed ripe bananas, ¼ cup milk
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Add this mixture to the wet ingredients and mix at low speed until just combined. Do not overmix!
2 cups all-purpose flour, 1 ½ teaspoons baking powder, 1 teaspoon baking soda, ¼ teaspoon salt
Pour the batter into the prepared pan and bake the cake in the preheated oven for 35-40 minutes or until a toothpick inserted into the center comes out clean. Remove from oven and let cool completely before frosting. Enjoy!
For the Frosting:
In a medium bowl, beat together the butter and peanut butter using a handheld electric mixer at high speed. Add the powdered sugar and vanilla and mix on low speed to combine. Lastly, add the milk and beat until smooth and creamy. Spread the frosting over top of the completely cooled cake in an even layer.
¼ cup salted butter, ½ cup creamy peanut butter, 1 cup powdered sugar, 3-5 tablespoons milk, ½ teaspoon vanilla extract
Notes
If using unsalted butter, simply add ¼ teaspoon of additional salt to the batter and ⅛ teaspoon of salt to the frosting.
I have only tested this frosting recipe with Kraft brand creamy-style peanut butter so I'm not sure how it would work using natural-style peanut butter.