Step 1: Preheat your oven to 350°F (175°C). Grease and flour a 13×18 inch (half-sheet) baking pan.
Step 2: In a large saucepan, combine the water (or coffee), 1 cup of butter, and 1/2 cup cocoa powder. Bring to a boil over medium heat, stirring occasionally. Once boiling, remove from heat.
Step 3: In a large bowl, whisk together the granulated sugar, all-purpose flour, baking soda, and salt.
Step 4: Pour the hot cocoa mixture from the saucepan into the dry ingredients and stir until just combined. Do not overmix.
Step 5: In a separate small bowl, whisk together the buttermilk, eggs, and 1 teaspoon vanilla extract. Add this mixture to the chocolate batter and mix until smooth.
Step 6: Pour the batter evenly into the prepared baking pan.
Step 7: Bake for 18-22 minutes, or until a wooden skewer inserted into the center comes out with moist crumbs. Do not overbake! While the cake is baking, prepare the frosting.
Step 8: For the frosting: In a large saucepan, melt 1/2 cup butter over medium heat.
Step 9: Stir in 1/4 cup cocoa powder, 1/3 cup milk, and 1 teaspoon vanilla extract. Bring to a boil, stirring constantly.
Step 10: Remove from heat and immediately whisk in the powdered sugar until smooth.
Step 11: Once the cake is out of the oven and still warm, immediately pour the hot frosting evenly over the entire surface of the cake. Spread gently if needed to cover. Allow the cake to cool completely before slicing and serving. This allows the frosting to set and become fudgy.