Brown the butter. Melt the butter in a light-colored pan over medium heat. Stirring constantly, continue to cook for 5 to 10 minutes, or until the butter smells nutty and the milk solids at the bottom of the pan turn golden brown. The butter will foam around the edges in the first few minutes of cooking. The foam will subside slightly before the butter is fully browned.

Immediately pour the browned butter into a large heatproof bowl and allow to cool slightly.

Preheat the oven to 350°F. Grease an 8x8-inch baking dish and line it with parchment paper.

To the butter, add the brown sugar, eggs, egg yolk, and vanilla and whisk until well combined. Sprinkle the salt over the mixture and whisk to combine, then add the flour and use a spatula to mix until just combined. Fold in the walnuts and white chocolate chips.

Transfer the batter to the prepared pan and use a spatula to spread it in an even layer. Press more walnuts and white chocolate chips into the tops of the bars and bake for 28 to 30 minutes, or until the edges are golden brown and small cracks form on top.

Allow to cool completely in the pan before slicing and serving.