1. Preheat oven to 350°F and prepare a 12-cup muffin pan with non-stick spray or paper liners.

2. Mix the brown sugar and cinnamon together in a small bowl and set aside.

3. In a large bowl, beat the granulated sugar and softened butter until light and creamy.

4. Add the eggs one at a time, mixing well after each addition, then stir in the vanilla extract.

5. In a separate bowl, combine the flour and baking powder.

6. Gradually add the dry ingredients to the butter mixture and stir in the milk until the batter is smooth.

7. Toss the chopped apples with 2 tablespoons granulated sugar and 1 teaspoon cinnamon.

8. Fill each muffin cup about halfway with batter.

9. Top with some of the apple mixture and sprinkle with the brown sugar-cinnamon mixture, gently pressing into the batter.

10. Add the remaining batter over the apples and spread evenly.

11. Top with the remaining apples and lightly press them into the batter.

12. If desired, use a knife or spoon to lightly swirl the brown sugar mixture through the apples and batter.

13. Bake for 20-25 minutes, or until a toothpick inserted in the center comes out clean.

14. Allow the muffins to cool for 15 minutes.

15. Whisk together the powdered sugar and milk or cream until the desired glaze consistency is reached.

16. Drizzle the glaze over the warm muffins and serve.