Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and light brown sugar together until light and fluffy, about 3 to 4 minutes.
Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Add half of the dry ingredients to the butter mixture and mix until just combined.
Stir in the sour cream until incorporated.
Add the remaining dry ingredients and mix until just combined.
Fold in the chopped apples gently.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 40 to 45 minutes or until a toothpick inserted in the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 15 minutes.
Transfer the cake to a wire rack to cool completely before frosting.
To make the cream cheese frosting, beat the softened cream cheese and butter together in a medium bowl until smooth and creamy.
Gradually add the sifted powdered sugar, beating until smooth and fluffy.
Mix in the vanilla extract.
Once the cake is completely cool, spread the cream cheese frosting evenly over the top.
Slice and serve.