Preheat oven to 350 F and arrange rack in the middle.
In a medium saucepan over medium heat, brown the butter. As the butter melts, swirl it around or stir it to prevent burning. The butter will froth during the browning process. Once the froth has dissipated, the butter should be golden brown and ready to remove from heat. Transfer to a heat-resistant bowl as soon as done to prevent further cooking. Set aside and allow to cool to room temperature.
In a separate bowl, whisk together flour, salt, and baking powder.
Add brown sugar to cooled butter and stir until evenly distributed. Add egg and vanilla then whisk until smooth, about 1 minute.
Add flour mixture to butter mixture and stir until just incorporated.
Add chopped walnuts and stir until evenly distributed.
Pour batter into a lightly greased 9 x 13 baking dish and smooth out the batter with the back of a spoon. Place walnut halves on top.
Bake until the top is crispy and is beginning to crack, 25-30 minutes. Remove from oven and let cool completely before attempting to remove from pan.
Don't want to use walnuts? Try chocolate or toffee chips. It's hard to go wrong with this one.