Preheat the oven to 350°. Generously grease and flour two 9-inch round cake pans. Using an electric mixer, combine the flour, 1 1/2 cups granulated sugar, the baking powder and salt. Mix in 2 sticks butter at low speed until the mixture is crumbly.
In a medium bowl, whisk together the eggs, egg yolks, 1/2 cup cream, applesauce and vanilla. Mix into the flour-butter mixture at medium speed until smooth, about 1 minute.
Divide the batter between the prepared pans. Bake until springy to the touch and a toothpick inserted into the center comes out with moist crumbs, 25 to 30 minutes. Transfer to a rack to cool for 10 minutes. Run a knife around the edges of the cake to release the layers, invert onto the rack and let cool completely.
In a medium skillet, melt 3 tablespoons butter over medium-low heat. Add the apple slices and remaining 1/3 cup granulated sugar and cook, stirring often, until the apples are tender and the juice is syrupy, about 15 minutes. Stir in the remaining 3 tablespoons cream and simmer for 3 minutes. Remove from the heat and let cool.
Meanwhile, using an electric mixer, beat the remaining 2 sticks butter and the confectioners' sugar until light and creamy. With the mixer on low, slowly mix in the caramel sauce.
Place 1 cake layer on a cake plate; spread the apple filling evenly on top, then cover with the remaining cake layer. Using an offset spatula or butter knife, cover the top and sides of the cake with the caramel frosting, swirling the top to make a decorative pattern.