In a medium pot, over medium heat, melt butter. Add diced apples and stir well.
Add brown sugar, stir, and cover with a lid.
Cook, stirring occasionally, until the apples are soft.
Drain off the juices accumulated during cooking.
Stir in Dulce de Leche until evenly incorporated. Set aside to cool.
Preheat oven to 350. Grease and flour 2 9-inch round cake pans.
Grate carrots on a fine grater and set aside.
In a large mixing bowl, whisk together eggs, oil, and sugar. Whisk vigorously for about a minute.
Add flour and baking soda and gently whisk until completely incorporated.
Stir in carrots.
Fold in 1 cup of cooked caramel apples into the batter.
Separate the batter among the 2 prepared baking pans. (You can weigh them on a scale to make sure that the batter is divided equally.)
Bake for 35-40 minutes.
Take the cakes out of the oven and allow them to cool for about 15 minutes.
Gently run a greased butter knife along the side of the cake pan to help separate the cake from the pan. Gently tap the cake pans on the counter to loosed the cake a bit, tip the cake pans, and take the cakes out of the pans.
Let the cakes cool completely on a wire rack.
Make sure the cakes and the apples are cooled completely before frosting.
Beat butter and cream cheese with an electric mixer of high speed for a couple of minutes.
Turn down the speed to low and add powdered sugar and vanilla extract.
As the sugar gets mixed in, bring the speed back up to medium-high. Beat for another minute.
Turn off the mixer and scrape sides and bottom of the bowl. Turn the speed back up and beat for another minute or so.
Assembling the cake:
Fill a piping bag with cream cheese frosting.
To make your cakes straight, cut off the domes of both layers of the cake. It's easiest to use a large serrated bread knife.
Spread 1/4 cup of Dulce de Leche over the bottom layer.
Spread remaining caramel apples over the bottom layer of the cake.
Pipe a thin layer of frosting over the apples.
Add second layer of the cake upside down, cut part down onto frosting.
Decorate the top by spreading a very thin layer of the frosting over the top first and then pipe some frosting over the edge. You can add some crushed pecans in the middle.
Drizzle caramel topping all over the top of the cake.