Preheat the oven to 350°F. Spray a 13x9-inch pyrex pan with nonstick spray and set aside.
In a large mixing bowl, combine the sugars, oil and egg whites. Beat with a wooden spoon until well blended. Add the flours, oat bran, baking soda, and spices. Stir just until moistened. Gently stir in the shredded apples.
Spread the batter evenly in the prepared pan. Bake for 25 to 30 minutes, or until a toothpick inserted into the center comes out clean. Cool.
If desired, sift powdered sugar on top of the cake. Cut into small squares to serve.