Prep. Preheat the oven to 350°F (175°C) with the rack in the center. Prepare a 9x9 or 8x8 square pan with non-stick cooking spray.
Beat butter and sugar. Beat the softened butter and granulated sugar in an electric stand mixer until light and fluffy. Add the eggs and vanilla, and continue beating until the mixture is well combined.
Mix dry ingredients. In a separate mixing bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, allspice, and cloves.
Combine. Gradually add the dry ingredients to the butter mixture, alternating with the sourdough starter and sour cream. Scrape the bowl with a spatula to make sure everything is incorporated. The batter will be very thick.
Add apples. Fold in the diced apples.
Make the topping. In a small bowl, cut the cold butter into the flour until it resembles pea-sized crumbs. Stir in the brown sugar, cinnamon, and nutmeg.
Assemble. Spread the cake batter into the prepared pan. Sprinkle the topping over the batter.
Bake. Bake for 40-45 minutes or until a toothpick inserted into the center comes out clean. Allow the cake to cool in the pan for at least 15 minutes.
Serve. Serve the cake in squares directly from the pan.