Preheat oven to 350 degrees. Coat a 9 x 13 inch baking pan with nonstick cooking spray and line with parchment paper; set aside.
In a large bowl, whisk together the flour, sugar, brown sugar, baking soda, cinnamon, nutmeg, and salt; set aside.
In a medium bowl, whisk together the applesauce, oil, and eggs until well combined. Pour the wet ingredients into the dry ingredients and stir until well combined.
Fold in the diced apples and walnuts (if using) until the mixture is well combined. Pour the batter into the prepared baking dish.
In a small bowl, mix together the sugar and cinnamon. Sprinkle it evenly over the top of the batter.
Bake the cake on the center rack of the oven for about 45 minutes, or until a toothpick inserted in the middle of the cake comes out clean. Remove from oven and let cool on a wire rack, then cut into pieces and enjoy!
Notes
Apples: you can use whatever apples you have on hand. I prefer to use a combination of Granny Smith and Gala or Honeycrisp apples when baking. Storage: store leftover cake in an airtight container in the refrigerator for up to a week. You can also freeze for up to a month.