Preheat the oven to 350°F (175°C). Grease and flour two 8-inch round cake pans.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat 1/2 cup softened butter and granulated sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition, then mix in 1 teaspoon vanilla extract.
Stir in the mashed bananas.
Alternately add the flour mixture and buttermilk to the banana mixture, beginning and ending with the flour mixture. Mix until just combined.
Divide the batter evenly between the prepared cake pans and smooth the tops.
Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cakes to cool in the pans for 10 minutes, then turn out onto wire racks to cool completely.
To make the frosting, beat 1/4 cup softened butter until creamy. Gradually add powdered sugar and beat until smooth.
Add 1 teaspoon vanilla extract and 2 tablespoons heavy cream, then beat until light and fluffy.
Place one cake layer on a serving plate and spread frosting evenly over the top.
Place the second cake layer on top and frost the top and sides of the cake.
Refrigerate for at least 30 minutes before serving to set the frosting.