Melt the butter in a light-colored saucepan over medium heat, stirring constantly so it doesn't scorch. After about five to eight minutes, it will foam up and then turn golden with little toasted specks at the bottom. Pour it into a heat-safe bowl right away to stop the cooking.
Return the same pan to the stove without washing it those browned bits add flavor. Add your chopped apples, maple syrup, and cinnamon, then cook over medium heat for three to five minutes until just softened. Set the apples aside to cool slightly while you prep the batter.
In a large bowl, whisk the slightly cooled brown butter with the brown sugar until combined. Add the eggs one at a time, then stir in the vanilla extract. The mixture should look glossy and smooth, almost like a caramel sauce.
Add the flour, baking powder, salt, cinnamon, and nutmeg directly into the wet mixture. Stir gently with a spatula just until no dry streaks remain, overmixing here can make the bars tough instead of chewy.
Gently fold the cooked apples into the batter, then spread everything into a parchment-lined 9x13-inch pan. Bake at 350°F for about 35 minutes, until the top turns lightly golden and a toothpick comes out with a few moist crumbs.
While the bars cool, melt and brown a small amount of fresh butter the same way as before. Whisk it with confectioners' sugar, milk, and vanilla extract until smooth. Drizzle it over the cooled bars before slicing into squares.