Preheat your oven to 350°F (175°C) and line a 9x13 inch baking pan with parchment paper.
In a large bowl, cream together melted butter and brown sugar until smooth.
Beat in eggs one at a time, then stir in vanilla.
In a separate bowl, whisk together flour, baking powder, salt, cinnamon, and nutmeg.
Gradually mix the dry ingredients into the wet mixture.
Fold in diced apples, being careful not to overmix.
Pour the batter into the prepared pan and spread evenly.
Bake for 25-30 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs.