In a small pot, stir together diced apples, brown sugar, cornstarch, cinnamon, and water. Cover and bring to a boil on medium-high heat. Uncover and cook until the apples are tender and the excess liquid has evaporated, stirring often. Transfer the apple compote to a small bowl to cool completely.

Preheat oven to 350?F. Line a 12-hole muffin/cupcake pan with parchment cupcake liners and set aside.

In a small bowl, stir together the flour, baking powder, cinnamon, and salt. Set aside.

In a large mixing bowl, add the softened butter. Use an electric hand mixer to beat the butter until it becomes very creamy. Add the granulated sugar and beat until fluffy. Add one egg at a time, beating thoroughly in between each addition. Finish with beating in the vanilla extract.

Use a fine mesh sieve to sift in half of the dry mixture. Mix on low speed briefly, just until barely combined. Add half of the milk and mix until barely combined. Repeat adding the remaining flour, barely mixing, the remaining milk, and mixing until just combined and smooth.

Fill the cupcake liners with batter until about ? full.

Bake for about 20 minutes or until a toothpick inserted in the middle comes out clean. Let the cupcakes cool in the pan for 5-10 minutes before transferring them to a wire rack to cool completely while you make the frosting.

In a small mixing bowl, add softened butter. Use a hand mixer to beat the butter until it becomes very creamy. Add the cream cheese, powdered sugar, cinnamon, and vanilla extract. Beat on low speed until it becomes a smooth and thick frosting.

Transfer the frosting to a piping bag fitted with a piping tip coupler (or use two piping bags) and a small ribbon piping tip. Set aside.

Use a small melon baller or a knife to scoop out the middle of each cupcake, about halfway deep.

Fill the hole in each cupcake with apple compote. Add another spoonful of apple compote on top of each cupcake, flatten it slightly, and leave the edge of the cupcake empty.

Starting with a small ribbon piping tip, pipe two crisscrossing lines of frosting over the apple topping. Switch to an open star piping tip. Pipe a border around the whole cupcake, on top of the lines of frosting.