Position oven rack to the middle. Preheat oven to 350° F (177°C). Prepare three 8-inch round cake pans with parchment rounds, and grease the sides. Set aside.
In a large bowl, add the flour, baking powder, baking soda, salt, cinnamon and nutmeg. Whisk to combine. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl using a whisk or hand mixer), add the oil and both sugars. Mix on medium-high speed until combined.
Add the applesauce and sour cream and continue to mix until completely combined.
Scrape down the sides and bottom of the bowl. Lower the speed to medium-low and mix in the eggs, one at a time until just combined. Mix in the vanilla.
Add the dry ingredients in three additions, mixing on low speed after each addition until completely combined. Using a rubber spatula, fold in grated apples.
Divide batter equally between the three cake pans, a little over 2 and ½ cups per pan. Bake for 35-40 minutes or until a toothpick inserted into the center comes out with a few moist crumbs.
Remove cakes from oven and allow to cool in pans for about 10 minutes, then turn onto cooling racks to cool completely.
FOR THE APPLE FILLING:
In a medium saucepan, combine the diced apples, lemon juice, cinnamon, nutmeg, butter, brown sugar and water.
Cook over medium heat, stirring occasionally until apples are softened, about 5-8 minutes.
Remove from heat and pour filling into a jar or an airtight container and allow to cool completely.
FOR THE CRUMBLE TOPPING:
Whisk flour, brown sugar, baking powder, cinnamon, nutmeg and salt in a medium bowl.
Add melted butter and mix until lumps form. Mix in oats. Chill in the refrigerator for 15 minutes.
Meanwhile, preheat the oven to 375°F. Transfer mixture to a parchment-lined sheet tray and bake for 8 minutes.
Using a metal spatula, stir and toss crumble. Rotate tray and continue to bake until crumble is dark golden brown, about 8 minutes more. Set aside to cool.
FOR THE CARAMEL CREAM CHEESE FROSTING:
In the bowl of a stand mixer, beat the softened butter and cream cheese until pale and fluffy, about 5-6 minutes.
Scrape down the sides and bottom of the bowl. Add the powdered sugar, one cup at a time and beat on low speed after each cup until combined.
Once combined add salted caramel sauce and continue to beat until combined and smooth.
Increase speed to medium-high and beat for about 5-7 minutes, or until frosting is light and fluffy.
