To begin, line two muffin pans with papers. Set aside. Preheat your oven to 350 degrees.
In a bowl, combine 3 cups of all-purpose flour, 1 teaspoon each of baking soda and salt, 1/4 teaspoon baking powder, and 2 teaspoons of cinnamon. Stir together and set aside.
In your mixing bowl, beat 3 eggs.
Pour in 1/4 cup extra virgin olive oil.
3/4 cup sour cream.
1/2 cup of plain applesauce
and a cup each of sugar and brown sugar, along with 2 teaspoons of vanilla. Beat those until creamy and smooth.
Add in the flour mixture. Mix until smooth.
Core and dice 3 cups of Granny Smith apple.
Stir in the apples. Scoop the batter into the baking pans. Bake for 17-19 minutes or until they test done with a tooth pick.
Remove to a baking rack and cool completely.
Now for the icing. Melt 1/2 cup (1 stick) of butter in a saucepan.
Add in a cup of brown sugar and 1/3 cup half & half.
Stir these together over medium heat, until melted and it begins to boil.
Allow it to boil 1 minute, and then while stirring vigorously, add in a teaspoon of vanilla. Turn off the heat and stir for another minute.
Set aside 2 tablespoons of the caramel sauce.
Place the rest in your mixing bowl and allow it to cool to room temperature, stirring occasionally.
Mix in 3 cups of powdered sugar, mixing until smooth.
If it is too thick, add a couple tablespoons of milk.
Dip the cupcakes into the icing.
Drizzle with a little of the reserved caramel sauce. If the sauce is too thick, pop it in the microwave for 5 seconds to thin it.