Preheat the oven to 350 F. Lightly oil and flour 3 parchment-lined 8″ round cake pans and set aside.
In a large mixing bowl, sift together the flour, baking soda, baking powder, cinnamon, ginger, nutmeg, and salt. Set aside.
In a medium-sized mixing bowl, mix together the finely ground flax seeds with the water. Let stand for 3 minutes, or until slightly thickened. Add the melted soy margarine, brown sugar, and vinegar, and whisk until well combined. Add the wet ingredients and the apple pieces and grated apple to the dry ingredients. Mix until just combined. Pour the batter into the prepared cake pans and bake until a toothpick inserted into the center emerges clean, about 35 minutes. Allow the cakes to cool in the pans for about 20 minutes, then invert the cakes onto a wire cooling rack to cool completely.
Layer cakes with vegan cream cheese frosting and serve!