To begin, preheat your oven to 350 degrees. Line 2 muffin pans (24 cupcakes) with papers. Set aside.
In a bowl, combine the flour, baking soda, baking powder, salt, and cinnamon. Stir together and set aside.
In your mixing bowl beat the eggs. Add in the sugars, oil, applesauce, sour cream, and vanilla. Beat until smooth.
Add in the flour mixture and beat until smooth.
Stir in the diced apples.
Scoop the batter into the pans.
Bake for 17-19 minutes or until they test done with a toothpick.
Remove from the pan and cool on a rack.
For the Icing:
Melt the butter, brown sugar, and half & half together in a sauce pan. Bring the mixture to a boil, stirring continually. Boil for 1 minute.
While stirring vigorously, add in the vanilla.
Remove from heat and stir for another minute.
Set aside 2 tablespoons of the caramel in a dish.
Pour the remaining sauce into your mixing bowl to cool to room temperature. Stir occasionally.
When the sauce is cooled, mix in the powdered sugar and beat until smooth. If it is too thick, add the milk and beat.
Dip the cupcakes into the icing. Drizzle a little of the reserved caramel sauce on the tops. If the sauce is too thick to drizzle, pop it in the microwave for 5 seconds.