01 - Preheat oven to 350°F. Line an 8-inch square baking pan with parchment paper, leaving an overhang on two sides for easy removal.
02 - Melt butter in a small saucepan over medium heat; continue to cook, swirling occasionally, until the butter foams and develops a golden brown color with a nutty aroma, about 5–7 minutes. Remove from heat and let cool for 10 minutes.
03 - In a medium bowl, toss diced apple with lemon juice, granulated sugar, and ground cinnamon. Set aside.
04 - In a large bowl, whisk together packed light brown sugar, granulated sugar, and cooled brown butter until smooth. Add eggs and vanilla extract, whisking until fully combined.
05 - Sift in all-purpose flour, baking powder, salt, and ground cinnamon. Stir gently until just integrated; avoid overmixing.
06 - Fold the apple mixture and chopped nuts (if using) into the batter until evenly distributed.
07 - Transfer the batter to the prepared pan, smoothing the surface with a spatula. Bake at 350°F for 30 to 35 minutes, until the top is golden and a toothpick inserted in the center emerges with moist crumbs.
08 - Allow to cool completely in the pan on a wire rack. Once cooled, lift out using parchment overhang and cut into 16 squares.