Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together 2 cups flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Set aside.
In a large bowl, cream the softened 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 the dry ingredients to the wet ingredients in three additions, alternating with the sour cream, beginning and ending with the dry ingredients. Mix until just combined.
Fold the diced apples gently into the batter.
Pour the batter into the prepared cake pan and spread evenly.
In a small bowl, combine the brown sugar, 1/4 cup flour, and cold cubed butter. Use a pastry cutter or your fingers to mix until the mixture resembles coarse crumbs.
Sprinkle the crumb topping evenly over the batter.
Bake for 50 minutes or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 15 minutes, then transfer to a wire rack to cool completely before serving.