Preheat oven to 350°F and line a 9x13-inch pan with parchment paper.
Melt butter in a medium saucepan over medium heat, stirring frequently until it turns golden brown and smells nutty, about 5 minutes; remove from heat.
Whisk brown sugar into hot brown butter until combined, then let cool 5 minutes.
Whisk in eggs and vanilla, then stir in flour, baking powder, and salt until just combined.
Fold in chocolate chips or walnuts, then spread batter evenly into prepared pan.
Bake 22-26 minutes until top is golden and shiny and a toothpick comes out with moist crumbs.
Cool completely before cutting into bars.