Preheat oven to 350 degrees. Coat an 8x8 or 9x9 baking pan with nonstick cooking spray.
In the large bowl of a stand mixer, beat eggs on medium-high speed. Lower the speed and gradually add the sugars, oil and vanilla, mixing well after each addition. Add cake meal and blend well.
In a small bowl, toss the sliced apples in half of the the cinnamon sugar mixture. Reserve the other half for sprinkling the top of the cake.
Pour half of the batter into the prepared baking dish. Smooth batter using a rubber spatula, and arrange half of the apple slices over the batter in a single layer. Repeat with the remaining batter and apples. Sprinkle the remaining cinnamon sugar mixture evenly over the layer of apples.
Bake for 90 minutes, until the top is golden brown. Allow to cool in the pan before slicing into squares.