Heat oven to 350° degrees. Spray an 8-inch square pan with cooking spray and line with parchment paper.
In a large bowl, mix together flour, sugar, cinnamon, nutmeg, baking powder, baking soda and salt.
In medium bowl, mix bananas, zucchini, egg, vegetable oil and 1 teaspoon vanilla. Add wet ingredients to dry ingredients all at once; stir just until combined.
Spread batter into prepared pan. Bake for 35 to 40 minutes or until cake just begins to turn brown on top and toothpick inserted in center comes out clean. Cool cake completely in pan, about 1 hour.
Meanwhile, in a medium bowl, beat ingredients for frosting with an electric mixer on medium speed until smooth and creamy. Spread frosting evenly over top of cake. Cover and refrigerate any remaining cake.