Preheat the oven to 350F. Butter generously a 9 -by 2-inch round cake pan.
In a small bowl stir together well ½ stick of the butter, melted, and the brown sugar and spread the mixture evenly in the prepared pan. Arrange the banana slices over the brown sugar mixture.
Into a bowl sift together the flour, the baking powder, the salt. In another bowl with an electric mixer cream, the remaining one stick of butter, softened, with the sugar until the mixture is light and fluffy, add the eggs, one at the time, beating well after each addition. Add the vanilla. Add the flour mixture in batches alternately with the milk
Pour the batter into the pan, spreading it evenly, and bake the cake in the middle of the oven for 45 to 55 minutes or until a tester comes out clean. Let the cake cool in the pan on a rack for 15 minutes, run a thin knife around the edge, and invert the cake into a plate.