For the best results, use a mandoline to slice the apples paper-thin. Be sure to buy regular applesauce, not unsweetened, for the applesauce cake layers. To prevent unwanted crystallization in the caramel, do not stir until step 2.
1. For the salted caramel filling: Combine sugar, water, and corn syrup in medium saucepan. Bring to boil over medium-high heat and cook, without stirring, until mixture is amber-colored around edge of saucepan, 4 to 6 minutes. Reduce heat to low and continue to cook, swirling saucepan occasionally, until evenly dark amber, 2 to 5 minutes longer.
2. Off heat, carefully stir in cream, butter, vanilla, and salt (mixture will bubble and steam). Return saucepan to medium heat and cook, stirring frequently, until smooth. Carefully transfer caramel to bowl and let cool completely, about 2 hours. (Once cooled, caramel filling will be very thick.)
3. For the applesauce cake layers: Adjust oven rack to middle position and heat oven to 325 degrees. Grease two 9-inch round cake pans and line with parchment paper.
4. Whisk flour, baking soda, salt, cinnamon, nutmeg, and cloves together in medium bowl; set aside. Whisk applesauce, sugar, melted butter, cider, eggs, and vanilla together in large bowl. Add flour mixture to applesauce mixture and whisk until just combined.
5. Divide batter evenly between prepared pans and smooth tops with rubber spatula. Bake until toothpick inserted in center comes out clean, 35 to 40 minutes, rotating pans halfway through baking. Let cakes cool in pans on wire rack for 15 minutes. Remove cakes from pans, discarding parchment, and let cool completely on rack, about 2 hours.
6. For the caramel buttercream frosting: Using stand mixer fitted with whisk attachment, whip butter on medium-high speed until smooth, about 20 seconds. Reduce speed to low and slowly add sugar, cream, vanilla, and salt. Whip until sugar is moistened, about 1 minute. Scrape down bowl and add ½ cup caramel filling. Whip on medium-high speed until light and fluffy, about 4 minutes, scraping down bowl as needed.
7. Place 1 cake layer on plate or pedestal. Spread remaining ½ cup caramel filling evenly over top, leaving ½-inch border around edge. Spread ¾ cup frosting evenly over caramel, right to edge of cake. Top with second cake layer, pressing lightly to adhere, then spread remaining frosting over top and sides of cake. Refrigerate cake until frosting is firm, about 20 minutes.
8. For the apple rosettes: Toss apples, sugar, and lemon juice in bowl until all slices are coated; let sit until apple slices are flexible, about 5 minutes. Shingle 5 apple slices lengthwise in straight line on cutting board. Starting at 1 end, roll up slices to form rose shape; place rose in center of cake. Repeat, arranging apple roses close together, until you have 5-inch-wide bunch of roses atop cake. Arrange remaining apple slices, overlapping with flat side facing down, around bottom edge of cake. Serve.