Preheat oven: Preheat to 350°F (175°C). Grease and flour a round cake pan.
Mix dry ingredients: In a medium bowl, whisk together flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
Combine wet ingredients: In a large bowl, whisk the egg, granulated sugar, and brown sugar until smooth. Add applesauce, oil, and vanilla; mix well.
Combine batter: Add dry ingredients to wet and stir until just combined. Fold in chopped apples.
Bake: Pour batter into the prepared cake pan and smooth the top. Bake for 35–40 minutes or until a toothpick inserted in the center comes out clean.
Cool & serve: Let cake cool in the pan for 10–15 minutes, then transfer to a wire rack. Serve warm or at room temperature with a dusting of powdered sugar or drizzle of caramel.
