Preheat Oven and Prepare Pan:
Preheat your oven to 350°F (175°C). Grease a 9×13-inch baking pan and line it with parchment paper. Grease the parchment paper as well.
Make the Cake:
In a large bowl, beat egg yolks with 1/2 cup of granulated sugar until thick and pale yellow. Stir in the vanilla extract, applesauce, and vegetable oil.
In another bowl, sift together flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Gradually add the dry ingredients to the egg yolk mixture, mixing until smooth.
Fold in chopped apples and chopped walnuts (if using).
Whip Egg Whites:
In a clean bowl, beat egg whites until soft peaks form. Gradually add the remaining 1/4 cup of granulated sugar, beating until stiff peaks form.
Combine Batter:
Gently fold the beaten egg whites into the apple batter, a third at a time, until well combined and no streaks remain.
Bake the Cake:
Spread the batter evenly into the prepared baking pan, smoothing the top with a spatula.
Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and let cool completely in the pan on a wire rack.
Make the Cheesecake Filling:
In a medium bowl, beat the cream cheese with powdered sugar and vanilla extract until smooth and creamy.
In another bowl, whip the heavy cream until stiff peaks form.
Gently fold the whipped cream into the cream cheese mixture until fully combined.
Prepare the Caramel Sauce:
In a medium saucepan, heat granulated sugar over medium heat, stirring constantly with a high heat resistant rubber spatula or wooden spoon.
Sugar will form clumps and eventually melt into a thick brown, amber-colored liquid as you continue to stir. Be careful not to burn.
Once sugar is completely melted, immediately add the butter. Be careful in this step because the caramel will bubble rapidly when the butter is added.
Stir the butter into the caramel until it is completely melted, about 2-3 minutes. A whisk helps if you find the butter is separating from the sugar.
Now slowly stir in cream and mix until the has cooled and cooled to cool completely Prepare preferredStyle let stand completely.
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