In a medium saucepan, combine all ingredients, and stir to combine. Cover the saucepan, and cook over medium heat for 15 to 20 minutes, or until apples are soft. Allow to cool, then blend. Chill sauce to room temperature before adding to cake batter (see recipe below).
The Apple Cake:
Preheat the oven to 325-degrees. Butter three 8-inch round cake pans, line the bottoms with parchment paper, and butter the parchment. Dust the parchment with flour and knock out the excess flour.Sift the flour, baking soda, baking powder, salt, cinnamon, allspice, and cloves together into a large bowl. Set aside.In the bowl of a standing mixer fitted with the paddle attachment, beat the butter until creamy, about 4 minutes. Add the sugar and beat until light and fluffy, about 3 minutes. Add the egg and beat until combined.
Add the flour mixture to the mixer bowl in three parts, alternating with the applesauce, beginning and ending with the flour mixture. Scrape down the bowl, then mix on low speed for a few more seconds.
Divide the batter among the prepared pans and smooth the tops. Bake for 40 to 45 minutes, rotating the pans halfway through the baking time, until a toothpick inserted in the center of the cake comes out clean. Transfer the pans to a wire rack and cool for 20 minutes. Turn the cakes out onto the rack, remove the parchment, and let cool completely.
In a medium saucepan with high sides, combine the sugar and corn syrup with 1/2 cup water. Stir the mixture gently so you don’t slosh any of it up the sides of the pan. Turn the heat to medium-high and continue stirring until the sugar dissolves. Increase heat to high, stop stirring, and allow the mixture to boil. Once it begins to turn a rich caramel color (if you don’t want to eyeball it, take the caramel to 300 degrees F on a candy thermometer), remove it from the heat, add the butter and cream, and stir until combined.
You can save the caramel sauce, tightly covered, in the refrigerator for up to 1 week. Let it come to room temperature before using it on cakes, ice creams, or quick breads.
If you want a warm topping, heat the caramel sauce in short bursts in the microwave or in the top of a double boiler.
In a medium, heavy-bottomed saucepan, whisk the sugar and flour together. Add the milk and cream and cook over medium heat, whisking occasionally, until the mixture comes to a boil and has thickened, about 10 to 15 minutes.Transfer the mixture to the bowl of a standing mixer fitted with the paddle attachment. Beat on high speed until cool. Reduce the speed to low and add the butter and vanilla; mix until thoroughly incorporated. Increase the speed to medium-high and beat until the frosting is light and fluffy.Add 1?3 cup of the caramel and continue mixing until combined. If the frosting is too soft, put the bowl in the refrigerator to chill slightly, then beat again until it is the proper consistency.