Preheat oven to 350°F and lightly grease a 9-inch round cake pan.
In a medium bowl, mix together the eggs and oil. Add the cinnamon, white and brown sugars, and vanilla extract and mix until combined.
Add the flour, salt, baking soda, and baking powder, and mix until thoroughly incorporated. Fold in the apples and the nuts and pour batter into prepared pan.
Bake until a toothpick inserted into the center comes out clean, about 4