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 add the dry ingredients in three additions, alternating with the sour cream, beginning and ending with the dry ingredients. Mix until just combined.
Pour half of the batter into the prepared cake pan and spread evenly.
Spoon the entire can of apple pie filling evenly over the batter layer in the pan.
Carefully spread the remaining batter over the apple pie filling layer, smoothing the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and allow it to cool in the pan on a wire rack for 15 minutes.
Run a knife around the edges of the cake to loosen it from the pan, then invert onto a serving plate.
Let the cake cool completely before slicing and serving.