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, cream the softened unsalted butter and granulated sugar together until light and fluffy, about 3 to 4 minutes.
Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
Add half of the dry ingredients to the butter mixture and mix until just combined.
Add the sour cream and mix until incorporated.
Add the remaining dry ingredients and mix until just combined. Do not overmix.
Fold in the diced apples gently until evenly distributed.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
In a small bowl, combine the light brown sugar and melted butter. Sprinkle this mixture evenly over the top of the batter.
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 15 minutes.
Run a knife around the edges to loosen the cake, then transfer it to a wire rack to cool completely before serving.