1. Preheat the oven to 350 F and generously grease and flour two 9″ round cake pans. Set aside.
2. In a large mixing bowl, whisk together the flour, baking soda, baking powder, cinnamon, nutmeg and salt. In another bowl, whisk together the applesauce, yogurt, and brown sugar until no lumps remain. Add the eggs and combine thoroughly. Pour in the melted (cooled) butter, whisking the entire time.
3. Combine the wet ingredients with the dry, stirring gently just until all of the dry ingredients are incorporated. Fold in the diced apples. Divide the batter evenly between the two cake pans, and place in the oven for 30-35 minutes, or until a toothpick comes out clean. Mine took about 32 minutes. Remove the cake pans and allow the cakes to cool for about 10 minutes before gently inverting and cooling completely on a wire rack.
4. While the cake is baking, it’s a good time to prepare the salted caramel sauce (click on the link). When your cake has cooled completely, you’re ready to frost it. In a stand mixer fitted with the paddle attachment, or using a handheld electric mixer, beat the butter for about 2 minutes or until light and fluffy. Add the salted caramel sauce and beat for 30 seconds to a minute to combine. Add the vanilla extract, and 1 cup of the powdered sugar and slowly combine, increasing the speed as the sugar begins to become incorporated. Repeat with the remaining 2 cups of icing sugar, using more or less sugar to achieve the right consistency. Frost your cake!

Cake will keep at room temperature for 4-5 days. Cakes (before icing) would freeze very well, up to 2 months if wrapped tightly in plastic wrap.