Preheat oven to 350 degrees with rack in the lower third of oven. Spray a 9-inch square baking pan with nonstick cooking spray. Line the bottom of the pan with waxed paper, then spray the paper. Dust the entire pan with flour and shake out the excess.
In a large bowl, whisk together the flours, turbinado sugar, cinnamon, and salt; set aside.
In a medium bowl, whisk together molasses, eggs, oil, and buttermilk.
Dissolve baking soda in the cup of boiling water. Fold the baking soda and molasses mixtures into dry ingredients until combined. (Do not overmix.) Fold in the minced ginger.
Scrape batter into the prepared baking pan; bake until cake is set around edges and a toothpick inserted in center comes out clean, about 40 minutes. Cool 10 minutes in the pan, then invert onto a cooling rack. Serve warm or at room temperature, dusted with confectioners' sugar.