Heat oven to 375 F.
Peel, core, and quarter the apples. Toss with a little salt and, unless they are very sweet, a bit of sugar to taste. If they are tart enough to make you squint, add the full measure of sugar. Spread on a rimmed baking sheet in a single layer. Cover with aluminum foil, and bake until the apples are soft, about 30 minutes. Mash with a potato masher or large fork. Allow to cool before proceeding.
For the Cake:
Heat the oven to 350° F and butter and flour a 10-inch springform pan. Sift together the flour, baking soda, salt, pepper and spices and set aside. In a large mixing bowl or the bowl of a standing mixer, beat the eggs with both sugars until light. Mix in the applesauce, oil and vanilla until smooth.
Using a spatula, fold in the dry ingredients, being careful not to over-mix. Pour the batter into the prepared pan and bake for about 45 minutes, until a cake tester inserted in the center of the cake comes out clean. Cool the cake for 10 minutes in the pan on a rack before turning it out and cooling completely on the rack—make sure the cake is not at all warm before you make the frosting.
For the Frosting:
In the bowl of an electric mixer fitted with the paddle attachment, cream the butter and confectioners' sugar until light and fluffy, 4 to 5 minutes.
Add the caramel sauce and mix on medium speed until incorporated. Scrape the bowl well.
Add the cream and vanilla and mix to combine. The frosting should be smooth and very soft. Scoop the frosting onto the cooled cake, and spread into an even layer. Garnish with flaky salt just before serving.