Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Set aside.
In a large bowl, cream the softened butter and granulated sugar together until light and fluffy, about 3-4 minutes.
Add the eggs one at a time to the butter mixture, beating well after each addition.
Stir in the vanilla extract.
Alternately add the dry ingredients and the applesauce to the butter mixture, beginning and ending with the dry ingredients. Mix until just combined.
Fold in the diced apples and chopped walnuts, if using.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 45-50 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.
Turn the cake out onto a wire rack and let it cool completely before serving.