Preheat the oven to 350 degrees Fahrenheit and line a large baking pan such as a 12x16 sheet pan with sides.
Heat the butter, water and cocoa powder in a medium saucepan over medium and let it come to a simmer for 5 minutes before removing the pan from the heat and letting cool completely for 20 minutes.
Mix the eggs, brown sugar and granulated sugar together in a large bowl until combined then stir in the sour cream.
Whisk the flour, baking soda and salt together in a medium size bowl then gradually add it into the wet ingredients until smooth.
Pour the batter into the baking pan and bake for 15-20 minutes or until a toothpick comes out clean when inserted into the center.
In the meantime, make the icing by adding the butter, milk, cocoa powder and vanilla to another saucepan over medium heat then let it simmer for about 10 minutes.
Gradually stir in the powdered sugar and when the cake is fresh out of the oven, evenly pour the icing over the cake and let it cool for 1 hour before serving.