Butter and flour 2 (9-inch) cake pans. Cut 2 circles of parchment paper or waxed paper to fit the bottoms of the pans, then press them inches In a mixer fitted with a whisk attachment (or using a hand mixer), cream the butter and shortening together. Add the sugar and mix until light and fluffy. One by one, add the eggs, mixing after each addition. With the mixer running at low speed, add the vanilla, cocoa, baking powder, baking soda, and salt and mix. With the mixer still running at low speed, add about 1/3 of the cake flour, then about 1/3 of the milk, and mix. Repeat with the remaining cake flour and milk and mix. Pour into the prepared pans and bake until dry and springy to the touch and a tester inserted into the center comes out clean (a few crumbs are okay), 30 to 35 minutes. Let cool in the pan for 15 minutes, then turn out onto wire racks and let cool completely, to room temperature. Using a long serrated knife, cut the cake layers in half horizontally. Reserving 3 halves for the cake, put the remaining half in a food processor, breaking it up with your hands. Pulse into fine crumbs.
Custard: Pour 2 1/2 cups of the water, the sugar, corn syrup and cocoa powder into a large non-reactive saucepan and bring to a boil over medium-high heat, whisking occasionally. Meanwhile, in a small bowl, whisk the remaining 1/2 cup of water and the cornstarch. Whisk into the cocoa mixture in the saucepan and return the mixture to a boil, whisking constantly. Cook, whisking constantly, until very thick, 3 to 4 minutes. Remove from the heat and stir in the butter and vanilla. Pour into a bowl. Cover with plastic wrap, lightly pressing the plastic against the surface to prevent a skin from forming. Chill until firm, about 45 minutes.
To finish the cake, place a cake layer on a cake plate or serving platter (reserving the most even layer for the top) and spread with cooled custard. Top with another layer of cake, then custard, then the final layer of cake. Cover the top and sides of the cake with the remaining custard. Coat the cake with the cake crumbs. Chill until ready to serve, at least 2 hours. Serve the same day.