Preheat oven to 325 F. Butter and flour a 9x13 baking pan.
In a medium bowl, whisk flour, baking soda, cinnamon, salt and nutmeg together. Set aside.
In a separate bowl, whisk eggs until combined. Add oil, sugars and vanilla and mix until well-combined.
Gradually whisk flour mixture into egg-sugar mixture just until blended.
Peel apples; cut into 1 1/2 inch chunks. Fold apples into batter. The batter will be very thick. Turn batter into prepared pan, spreading to edges. Bake for 1 hour or until toothpick inserted in center of cake comes out clean.
Remove cake from oven. Cool on wire rack for 30 minutes while preparing caramel glaze.