Preheat the oven to 350°F. Grease a 9-inch round baking pan and line the bottom with parchment paper.
In a large bowl, whisk together the brown sugar, granulated sugar, and eggs until pale and foamy, about 1 minute. Add the oil, vanilla, cinnamon, nutmeg, and salt. Whisk until smooth.
Add the flour, baking powder, and baking soda to the wet mixture. Whisk until well combined.
Gently fold in the chopped apples using a rubber spatula, ensuring they are evenly distributed throughout the batter.
Pour the batter into the prepared pan. Smooth the top with an offset spatula. If using, sprinkle on chopped nuts and gently press them into the batter.
Bake for 30-35 minutes, or until a tester inserted into the center comes out clean.
Allow the cake to cool in the pan on a wire rack for about 15 minutes. Run a knife around the edge of the cake, then turn it out onto the rack to cool completely.