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, ground cinnamon, ground nutmeg, and salt. Set aside.
In a large bowl, beat the softened unsalted 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 to the butter mixture, beginning and ending with the flour mixture. Mix just until combined.
Fold in the diced apples gently with a spatula.
Pour the batter into the prepared cake pan and spread evenly.
Bake in the preheated oven for 45 to 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.
Run a knife around the edges to loosen the cake, then invert onto a serving plate.
Spread the apple pie filling evenly over the top of the cooled cake.
Serve the cake warm or at room temperature.