Make the Cake: Preheat oven to 375 degrees F. Grease a 9x13-inch baking pan with non-stick cooking spray.
In a medium saucepan, stir together the sugar, unsweetened chocolate, butter and water. Place over medium heat and stir occasionally until all of the ingredients are completely melted together. Remove from the heat and let cool slightly, about 10 minutes.
Whisk in the buttermilk and egg. Sift the flour, baking soda, baking powder and salt into the pot in three additions, whisking gently to combine after each addition. Add the vanilla and whisk briefly to blend.
Pour the batter into the prepared pan and spread into an even layer. Bake until a skewer inserted in the center comes out clean, about 30 to 40 minutes. Allow the cake to cool completely before frosting.
Make the Frosting: Using an electric mixer, beat together the cream cheese and butter on medium-high speed until creamy, about 3 minutes, scraping down the bowl as needed. Sift in the powdered sugar and cocoa powder, then add the salt. Mix on low speed until well combined, scrape down the bowl, then increase to medium-high speed and beat until smooth and whipped, about 1 minute. Add the vanilla extract and beat on medium-high for an additional 1 minute.
Spread the frosting evenly over the top of the cake. The cake can be stored, covered, at room temperature or in the refrigerator for up to 4 days.