In a small bowl or inside the mug, combine the melted butter, brown sugar, vanilla, milk, flour, baking powder, cinnamon, and nutmeg. Stir or whisk to combine the ingredients until smooth. Fold in the chopped apples.
Microwave the mug cake for 1 to 1½ minutes. In my microwave, it's best at 1 minute 15 seconds. Begin checking at 1 minute. The cake should look set, but be careful not to over-microwave.
Let cool before topping with whipped cream or ice cream, caramel sauce, or dust with powdered sugar.