Preheat the oven to 350°F (175°C). Line an 8-inch square baking pan with parchment, leaving overhang on two sides for a sling.
Melt the butter and let it cool until just warm to the touch, about 5 minutes.
In a medium bowl, whisk the brown sugar into the melted butter until glossy and combined, 30–60 seconds.
Whisk in the egg and vanilla until the mixture is thick, smooth, and lightened slightly, about 30 seconds.
In a separate small bowl, whisk together the flour, baking powder, and salt.
Add the dry ingredients to the wet ingredients and stir with a spatula just until no dry flour remains; do not overmix.
Fold in the chopped walnuts until evenly distributed.
Scrape the batter into the prepared pan and smooth the top.
Bake until the edges are set and lightly golden and a toothpick inserted near the center comes out with a few moist crumbs (not wet batter), 20–25 minutes.
Cool the pan on a rack for 20–30 minutes. Use the parchment sling to lift the slab out, cool 15 minutes more, then cut into 16 squares and serve.