Preheat the oven to 350 degrees F. Line the bottom of an 8x8 baking dish with parchment paper.
Make the brown butter: In a small pan set over medium heat, melt the butter and cook until it turns brown and smells nutty, about 5 minutes. Try not to walk away from the butter, as it can go from brown to burned very quickly. Pour the brown butter into a large mixing bowl and let it cool slightly.

Whisk together the flour, baking powder, and salt to combine.

Add both sugars to the brown butter, and mix on medium speed with a hand mixer until well combined, about 2 minutes. Add the eggs and beat for another 2 minutes, until the mixture has lightened in colour. Add the vanilla, and beat for 30 seconds.
With the mixer on low, slowly add the flour mixture, and stop mixing right before the flour disappears.

Add the chopped walnuts, then finish stirring by hand with a spatula, just until the flour is fully mixed in.
Pour the mixture into the prepared pan and bake for about 35-40 minutes, until a toothpick inserted into the center comes out clean. Cool on a wire rack, and cut into squares. Enjoy!