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, salt, cinnamon, and nutmeg. Set aside.
In a large bowl, beat the softened butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Reduce the mixer speed to low and add the dry ingredients in three additions, alternating with the sour cream, beginning and ending with the dry ingredients. Mix until just combined.
Fold in the chopped apples gently with a spatula.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 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 15 minutes. Then transfer the cake to a wire rack to cool completely.
While the cake cools, prepare the cream cheese frosting. In a medium bowl, beat the softened cream cheese and butter together until smooth and creamy.
Gradually add the powdered sugar, beating on low speed until incorporated. Add the vanilla extract and beat on medium speed until smooth and fluffy.
Once the cake is completely cool, spread the cream cheese frosting evenly over the top.
Slice and serve.