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MAKE THE CAKE: Preheat the oven to 375°. Butter two 9-inch* round cake pans and coat lightly with flour. Line the bottoms with parchment paper. In a standing electric mixer fitted with the whisk, beat the 1 stick of butter with the shortening until creamy. Add the sugar and beat at medium speed until fluffy, about 3 minutes. Add the eggs one at a time, beating well between additions. Add the vanilla. At very low speed, beat in the cocoa powder, baking powder, baking soda and salt. Add the 2 1/4 cups of cake flour and the milk in 3 separate alternating batches, scraping down the side and bottom of the bowl occasionally.
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Divide the cake batter between the prepared pans and smooth the tops. Bake in the center of the oven for about 30 minutes, until a toothpick inserted in the centers comes out with a few moist crumbs attached. Let the cakes cool for 15 minutes, then invert them onto a rack and let cool completely.
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MEANWHILE, MAKE THE FILLING In a large saucepan, combine 2 1/2 cups of the water with the sugar, corn syrup and cocoa powder and bring to a boil, whisking constantly. In a bowl, whisk the cornstarch with the remaining 1/2 cup of water until smooth; whisk into the cocoa mixture. Cook over moderately high heat, whisking constantly, until very thick, 3 minutes. Off the heat, whisk in the butter, vanilla and salt. Scrape the filling into a bowl and press a sheet of plastic wrap onto the surface of the filling. Let cool, then refrigerate until firm, 45 minutes.
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Using a serrated knife, halve each cake layer horizontally. Break up the less attractive top cake layer and transfer to a food processor; pulse into crumbs. Reserve the two cake bottoms and one smoother top.
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Set one of the cake bottoms on a cake plate and spread with 1 1/2 cups of the filling. Top with the second bottom layer and another 1 1/2 cups of filling. Cover with the cake top and spread the remaining filling all over the top and side. Pat the crumbs all over the cake. Refrigerate for at least 1 hour before serving.
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MAKE AHEAD: The chocolate cake can be refrigerated overnight.