Preheat the oven to 350°F (175°C). Grease and flour an 8-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 alternately add the flour mixture and sour cream in three additions, beginning and ending with the flour mixture. Mix until just combined.
Fold the diced apples gently into the batter using a spatula.
Pour the batter evenly into the prepared cake pan and smooth the top with a spatula.
Bake for 38 to 42 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Run a knife around the edges to loosen the cake, then invert it onto a wire rack to cool completely before using as a cake layer.