Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, beat the granulated sugar, light brown sugar, and eggs together until smooth and creamy.
Add the vegetable oil, vanilla extract, and unsweetened applesauce to the sugar and egg mixture. Mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the diced apples and chopped walnuts (if using) evenly into the batter.
Pour the batter into the prepared cake pan and spread it evenly.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.