In a large mixing bowl, beat together butter, oil, granulated sugar, and brown sugar. Beat for 3-4 minutes, until the mixture turns a lighter color and is fluffy.
1/4 cup unsalted butter, 1/4 cup vegetable oil, 1/4 cup granulated sugar, 1/2 cup brown sugar
Add vanilla and sour cream, mix together, then add one egg at a time and mix until just combined.
Add the milk slowly, mixing as you add it, then add in half the flour mixture, mix well. Add the remaining flour mixture and mix until fully incorporated, be careful not to over mix.
3/4 cup milk
Fold in apple chunks until evenly distributed throughout batter.
1 cup finely chopped apple
Fill cupcake liners 3/4 full of batter then bake for 15-18 minutes, or until a toothpick inserted in the center comes out clean.
Remove cupcakes from the pan and place on a baking rack to cool completely.
Caramel Buttercream Frosting
In a large mixing bowl, beat the butter until it is smooth and creamy.
1 1/4 cups unsalted butter
Add 1/2 of the powdered sugar, mix well, then add 4 tablespoons of caramel sauce and mix until combined.
5 cups powdered sugar
Add the remaining 1/2 of the powdered sugar and beat well.
Add the cinnamon and mix until frosting is a good piping consistency. If it is too thick, add more caramel sauce.
1/8 tsp cinnamon
Use a piping bag with a large piping tip to pipe frosting onto the cupcakes, then drizzle with caramel sauce, and top with a slice of apple.
Notes
Rub lemon juice on the apple slices to keep them from browning.For extra apple flavor, you can add an apple compote to the center of the cupcake.If you don’t have a pipping bag you can use a ziplock with the corner cut off.Store at room temperature in an airtight container for up to three days.Freeze the baked cupcakes in a freezer-safe container for up to three months, then unthaw and frost when ready to serve.