Preheat your oven to 350°F (175°C) and position the rack in the center. Grease and line a 9-inch springform pan with parchment paper, allowing some overhang for easy removal.
Peel, core, and dice your apples into roughly ½-inch cubes. Toss the diced apples with 1 tablespoon of lemon juice to prevent browning and enhance flavor.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a separate large bowl, cream the butter and sugars until light and fluffy (about 3 minutes with an electric mixer).
Add eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Gradually add the dry ingredients to the wet ingredients, alternating with the sour cream, beginning and ending with the dry ingredients.
Fold in two-thirds of your diced apples, saving the remainder for the middle layer.
Spread half of the batter into your prepared pan. Scatter the remaining diced apples over this layer, pressing them gently into the batter.
In a medium bowl, combine the flour, brown sugar, cinnamon, and salt for the crumble. Using your fingertips, work the cold cubed butter into the mixture until it resembles coarse crumbs with some pea-sized pieces remaining. Stir in the nuts if using.
Carefully spread the remaining cake batter over the apple layer, then generously sprinkle the crumble mixture over the top, covering the surface completely. Gently press the crumble into the batter slightly so it adheres during baking.
Bake for 50-55 minutes, or until a toothpick inserted into the center comes out clean with a few moist crumbs. If the top begins to brown too quickly, loosely cover with aluminum foil after about 35 minutes.
Allow the cake to cool in the pan on a wire rack for 20 minutes, then remove the sides of the springform pan. Let cool for another 20-30 minutes before slicing.