For the cake: Heat oven to 350° F. Lightly coat a 9-by-13-inch baking pan with vegetable cooking spray; set aside.
In a medium bowl, combine the cocoa, 1/4 cup granulated sugar, and 1/4 cup water. In a large bowl, with an electric mixer at medium-high, beat the butter with the remaining sugar until light and fluffy. Reduce speed to low, add the remaining ingredients, and beat until smooth.
Spoon 2 cups of the vanilla batter into the bowl with the cocoa mixture; blend until smooth. Pour the vanilla batter into baking pan. Spoon in the chocolate batter to create six puddles on top of the vanilla batter. Using a knife, drag through the two batters to marbleize. Bake 22 to 25 minutes or until a toothpick inserted in the center comes out clean. Remove from oven and cool completely on a wire rack.
For the frosting: Melt the chocolate in a Pyrex measuring cup in a pan of simmering water; stir.
In a medium bowl, with an electric mixer at medium speed, beat the butter, sugar, and melted chocolate, adding just enough milk to make a spreadable frosting. Frost the cooled cake.