Begin by placing 1 cup of unsalted butter in a saucepan over medium heat. Allow it to melt and then watch closely as it bubbles and begins to turn golden brown. Stir occasionally to ensure even cooking. Once it has a nutty aroma and is a rich amber color, remove it from the heat and let it cool slightly.
While your butter is cooling, preheat your oven to 350°F (175°C). Grease and flour your 9x13 inch baking pan, or line it with parchment paper for easy removal.
In a large mixing bowl, whisk together the 2 cups of all-purpose flour, 1 teaspoon of baking powder, and 1 teaspoon of salt. This mixture will help ensure an even distribution of the leavening agent.
In another bowl, combine the browned butter, 1 cup of water, 2 cups of granulated sugar, 2 large eggs, 1 cup of sour cream, and 1 tablespoon of vanilla extract. Use an electric mixer to beat the ingredients until smooth and well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Be careful not to overmix, as this can lead to a dense cake.
Pour the batter into the prepared baking pan and smooth the top with a spatula. Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, remove the cake from the oven and allow it to cool in the pan for about 10 minutes. Then transfer it to a cooling rack to cool completely.
In a mixing bowl, beat together 6 tablespoons of unsalted butter until creamy. Gradually add the 6 cups of powdered sugar, 1 teaspoon of vanilla extract, and 8 tablespoons of milk. Mix until smooth and spreadable. Add more milk if necessary to achieve your desired consistency.
Once the cake is completely cooled, generously spread the frosting on top using a spatula. You can leave it rustic or create a smooth finish, depending on your preference.
Slice the cake into squares and serve it up! This Browned Butter Sheet Cake is delightful on its own, or you can pair it with fresh berries or a scoop of ice cream for an extra treat.
