Preheat oven to 325°F and grease two 8-inch round cake pans or an 8×8 baking dish with cooking spray.
In a large mixing bowl, prepare the flax eggs by mixing flaxseed meal with water. Allow them to sit for about 5 minutes.
Combine sugar, molasses, oil, vanilla, milk and grated apples with the flax eggs. Whisk well.
Add all remaining ingredients and stir until combined. Pour batter into prepared pans.
Bake for about 30-35 minutes if using two round cake pans or40-50 minutes for an 8×8 dish. A toothpick inserted into the center should come out clean, and the edges should be visibly browned.
Remove from oven and allow to cool before removing the cake from the pan.