Preheat the oven to 350 degrees Fahrenheit.
In a large bowl, whisk together the brown sugar, white sugar, oil, and applesauce until well combined.
Add in the eggs, and stir until just smooth.
In a separate large bowl, combine your flour, baking soda, baking powder, salt, cinnamon, and nutmeg. Whisk them together.
Add half the dry ingredients into the wet ingredients, while mixing until just combined.
Add in half the buttermilk and stir till just combined.
Repeat with the remaining dry ingredients, then buttermilk.
Fold in the apple pieces.
Pour the batter into a greased 9x13inch baking pan.
Bake for 30-35 mintues, or until an inserted toothpick comes out clean (a few crumbs are okay)
Let the cake cool completely, then spread the frosting over the top.