Preheat oven to 350°F, and place a rack in the center. Grease a 7-in round springform cake pan with vegetable oil.
In a large bowl, whisk together the flours, sugars, baking soda, salt, fresh ginger, and ground cinnamon.
In a separate bowl, mix together the olive oil, eggs, mashed banana, yogurt, and vanilla. Pour the banana mixture into the flour mixture and fold with a spatula until just combined. Add the nuts if using. Scrape the batter into the prepared pan and bake until golden brown, about 45 to 50 minutes.
Transfer the pan to a wire rack to cool in the pan for 10 minutes, then turn the cake out of the pan to cool completely. Once cool, carefully cut in half.
Make Frosting
Place the margarine in a large bowl and beat on high speed until creamy. Beat in the sugar and the lemon zest. Add the lemon juice, adding a few drops more if necessary. Fill and frost the top and sides of the cake with the icing.