Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius). Grease and flour two 9-inch round cake pans.
In a large bowl, whisk together flour, baking soda, and salt.
In a separate bowl, cream together butter, granulated sugar, and brown sugar until light and fluffy.
Beat in eggs one at a time, then stir in vanilla extract and mashed bananas.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Divide the batter evenly between the prepared cake pans.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Let cakes cool in pans for 10 minutes, then transfer to a wire rack to cool completely.
While the cakes are cooling, make the frosting. In a large bowl, cream together butter and peanut butter until smooth.
Gradually add powdered sugar, alternating with milk, beating until smooth and creamy. Stir in vanilla extract.
Once cakes are cool, frost one layer, then place the second layer on top and frost the entire cake.
Notes
Use ripe bananas for the best flavor. The darker the banana peels, the sweeter the cake will be.
Do not overmix the batter, as it can make the cake tough.
If you prefer a less sweet frosting, you can reduce the amount of powdered sugar.
You can store the cake in an airtight container at room temperature for up to 3 days.