Heat the oven to 350°F (175°C). Line the pan with a parchment sling. Measure all ingredients before browning the butter.
Melt butter in a light-colored saucepan over medium heat. Stir as it foams; remove when the milk solids turn golden brown and smell nutty.
Immediately scrape butter and brown flecks into a heatproof mixing bowl. Cool for 10–15 minutes until warm rather than hot.
Whisk brown sugar into the butter. Add egg, yolk, and vanilla, mixing until the batter looks glossy and evenly combined.
Whisk flour, baking powder, and salt separately. Fold into the butter mixture, then fold in optional white chocolate chips.
Spread in the prepared pan. Bake 23–28 minutes, until the edges are set and a tester has moist crumbs rather than raw batter.
Cool completely in the pan on a rack. Avoid pressing the center while hot because the soft crumb needs time to firm.
Lift out and cut into 16 squares. Use a sharp knife, wiping between cuts, and store only after the bars have cooled fully.
