Cake: Preheat oven to 350 degrees. Butter a 9-by-13-inch baking dish; line bottom and 2 long sides with parchment, leaving an overhang. Butter parchment, and dust with cocoa powder. Whisk together cocoa powder, flour, granulated sugar, baking soda, baking powder, and salt in a large bowl. Gather flour mixture into a mound, and create a well in center. Pour eggs, buttermilk, coffee, oil, and vanilla into well, and whisk until thoroughly combined and smooth. Pour batter into dish, and spread evenly with an offset spatula. Bake until set and a toothpick inserted into center comes out clean, about 25 minutes.
Topping: Meanwhile, melt butter for topping in a small saucepan. Combine confectioners' sugar, cocoa powder, and coconut in a large bowl. Stir coffee and vanilla into butter, and pour over coconut mixture. Stir to combine, and immediately spread evenly over warm cake. Let cool completely on wire rack.
To serve, use parchment to transfer cake to a cutting board. Cut into thirds lengthwise, then cut into quarters crosswise (you should have 12 pieces). Serve with whipped cream and toasted coconut.
Cake can be refrigerated in dish, wrapped in plastic wrap (do not let plastic touch cake), up to 3 days.