Banana Cake with Coconut Frosting

Put oven rack in middle position and preheat oven to 350 °F. Lightly butter and flour a 9-inch round cake pan, knocking out excess flour. Whisk together flour (1 1/2 cups), baking powder, baking soda, and salt in a small bowl. Beat together butter (1 stick) and brown sugar in a large bowl with an electric mixer at high speed until pale and fluffy. Beat in egg until combined, then add bananas, sour cream, and vanilla and beat until combined well. Reduce speed to low and add flour mixture, mixing until just incorporated. Spread batter in cake pan and bake until pale golden and a wooden pick or skewer inserted in center comes out clean, 20 to 25 minutes. Cool in pan on a rack 10 minutes. Run a thin knife around edge of pan, then invert rack over cake and invert cake onto rack to cool completely.