Cake: Preheat oven to 350°F (conventional oven). Grease a tube pan or Angel Food Cake pan.
In a stand mixer, cream the sugar, butter, and vanilla until smooth. Add eggs one at a time, mixing well after each addition.
In a separate bowl, whisk together the cake flour, sifted cocoa powder, baking soda, salt, and instant chocolate pudding mix.
Gradually add the dry mixture and buttermilk to the wet ingredients, alternating between the two, until fully incorporated.
Stir in the semi-sweet chocolate chips by hand using a wooden spoon or rubber spatula.
Pour the batter into the prepared tube pan.
Bake at 350°F for 30 minutes, then reduce heat to 325°F and bake an additional 35-45 minutes, or until a cake tester or skewer inserted comes out clean.
Let the cake cool completely on a wire rack before inverting onto a cake plate for frosting.
Chocolate Icing: In a medium saucepan, combine water, butter, and vanilla. Heat until butter is melted.
Remove from heat and stir in the sifted cocoa powder, followed by the sifted powdered sugar. The mixture will be thick.
Stir in heavy cream, one tablespoon at a time, until the icing is smooth and pourable.
Pour the icing over the cooled cake, starting from the center and letting it drizzle over the sides. The frosting will firm up as it cools.