Preheat the oven to 350 F, and position a rack in the center of the oven. Line a 9-inch square baking dish with parchment paper, or spray it with non-stick baking spray.
In a saucepan, melt the butter over medium heat. Continue to cook the butter until nutty, golden brown milk solids form at the bottom of the pan. This process might take about 10 minutes. First the butter will hiss and splatter as the water evaporates, then as it quiets down, a layer of foam will form on top of the butter. This means the butter is done. Immediately remove from the heat, and scrape the butter, including the brown milk solids, into a large bowl. Let cool for 10 minutes.
Whisk the espresso powder into the butter, then add the brown sugar, granulated sugar, eggs, vanilla and milk, and whisk until combined. If some of the butter solidifies into small pieces, this is nothing to worry about.
In a separate bowl, sift together the flour, salt and baking powder, then whisk until everything is evenly distributed.
Add the dry ingredients to the wet, whisking until moistened, then fold in the chopped walnuts.
Pour the batter into the prepared pan, and bake for about 30-32 minutes, until the top springs back when gently touched, or a toothpick or cake tester comes out clean. Set the pan on a wire rack, covered with a clean kitchen towel, and cool completely before frosting.