In a medium, heavy-bottom saucepan, combine brown sugar, butter, and milk. Heat over medium, whisking occasionally until butter is melted and sugar is dissolved. Stop stirring and cook until mixture begins to simmer. Remove from heat and cool to room temperature, at least 1/2 hour.
Lightly grease 12 cups in a mini muffin tin. Preheat your oven to 400°F.
Combine diced apples, butter, granulated sugar, brown sugar, vanilla, cinnamon, nutmeg, and salt in a medium saucepan. Bring to a simmer over medium heat and cook for 4 to 6 minutes until apples are softened.
On a well-floured surface, roll out dough to 1/8-inch thick. Cut out 12 1/4-inch rounds and 12 7/8-inch rounds. Press larger rounds into greased muffin tins. Fill pie crust with apples. Top with the smaller rounds. Use a fork to crimp down slightly around the edges, and cut an x into the tops for steam to escape.
Bake for 15 to 18 minutes until lightly browned and set aside to cool. Use a plastic knife to pop them out of the baking tray if any of them stick.
Turn oven temperature down to 350°F. Line 12 regular-size cupcake cups with cupcake liners.
In the bowl of your stand mixer or in a large bowl if using a handheld electric mixer, whisk together flour, sugar, baking powder, and salt. Add butter, sour cream, eggs, and vanilla and beat on medium for 30 seconds to one minute until very smooth, scraping the sides and bottom of the bowl at least once halfway through.
Add about 1 heaping tablespoon of batter to the bottom of each of the cupcake cups. Place a mini pie in each of the cups and press down until it almost touches the bottom. Fill the cups with just enough batter to cover the top of the mini pies and smooth over. You will not use all of the cupcake batter. Do not overfill!
Bake for 15 to 18 minutes until cupcakes are set and the edges spring back when lightly pressed.
Cool completely before frosting.
Transfer cooled caramel to the bowl of your stand mixer add 1/2 of the powdered sugar. With the paddle attachment, beat caramel on medium until smooth and slowly add in the rest of the sugar until frosting reaches your desired consistency.
Pipe or spread frosting on cupcakes. (If not piping, cut frosting in half!)
Serve and enjoy!