Add sugar and beat until light and fluffy, 3 minutes. Add eggs and beat until combined. Reduce speed to low and add flour mixture to mixer bowl in three parts, alternating with applesauce, beginning and ending with flour mixture. Scrape sides of bowl and mix on low an additional 30 seconds to combine.

Divide batter among prepared pans. Bake until a toothpick inserted in center comes out clean, 45-55 minutes, rotating pans halfway through baking time. Transfer pans to wire rack and cool 20 minutes. Turn cakes onto rack, remove parchment paper, and cool completely.

Salted Caramel Buttercream In a medium saucepan, whisk sugar and flour. Add milk and cream and cook over medium heat, whisking occasionally, until mixtures comes to a boil and has thickened, 10-15 minutes. Transfer to bowl of a standing mixer fitted with paddle attachment. Beat on high speed until cool, about 5 minutes.

Reduce speed to low, add butter and vanilla; mix until thoroughly incorporated. Increase speed to medium-high and beat until frosting is light and fluffy. Add Salted Caramel Sauce and continue mixing until combined and fluffy. If frosting is too soft, chill bowl slightly in refrigerator, then beat again until it reaches a fluffy consistency.

Assemble Cake If any of cake layers have domed on top, trim top to create flat surfaces. Place one cake layer on a serving platter. Spread 1¼ cups of frosting on top. Add next layer and top with 1¼ cups of frosting. Add third layer on top and spread a very thin layer of frosting over sides and top of cake. Refrigerate 15 minutes until firm. Frost sides and top of cake with remaining frosting. Drizzle additional Salted Caramel Sauce on top and sprinkle with chopped peanuts. Return to refrigerator at least 15 minutes prior to serving. Allow cake to sit at room temperature for at least 2 hours before serving.