Melt the butter over medium heat in a medium saucepan until melted. Turn the heat down to medium-low and continue to cook the butter until the fat turns golden brown, about 10 minutes. The fat will first form a white foam on the top of the pan, then sink to the bottom and brown. This is best done in a light-colored pan so that the browning process can be seen. You know the butter is properly browned when the fat solids look golden and the butter has a nutty scent. Remove browned butter from the heat immediately and cool in the refrigerator until the butter is cooled and has a creamy consistency, about 2 hours.
Preheat oven to 350ºF and butter and flour 2 loaf pans or 2 10" cake pans. Pound cakes are traditionally baked in a loaf shape, but I prefer to use cake pans as I find the cake is less likely to sink in the middle. Alternatively, line the pans with parchment paper and then grease the parchment. Add the browned and cooled butter to the bowl of a stand mixer fit with a paddle attachment, or to a large mixing bowl. With the mixer turned on low, or with a wooden spoon in the mixing bowl, beat the sugar and vanilla into the butter until they are creamed together.
In a separate medium mixing bowl, whisk together the flour, baking powder, and salt until combined. With the mixer on low speed, or in the mixing bowl with a wooden spoon, slowly add the flour, a little at a time, to the butter mixture, alternating with 1/4 cup of the milk, until all the flour and milk have been added to the batter. Turn the mixer on high, or beat vigorously with the spoon, for another 2 to 3 minutes or until the batter is completely smooth and thickened. Pour the batter evenly between the two prepared pans, smoothing the top with a spatula. Bake in the preheated oven for 45 to 50 minutes, or until the cakes are golden brown on top and a toothpick inserted in the center comes out clean. Cool completely on wire racks before removing from pans and serving.