Early in the day, preheat the oven to 350 degrees F and generously butter the bottom and sides of one 9-inch springform pan (preferably a nonstick one). Wrap the outside with aluminum foil, covering the bottom and extending all the way up the sides.
To make the cheesecake layer, put one package of the cream cheese, 1/3 cup of the sugar, and the cornstarch in a large bowl and beat with an electric mixer on low until creamy, about 3 minutes, scraping down the bowl a couple of times. Blend in the remaining cream cheese, one package at a time, scraping down the bowl after each. Increase the mixer speed to medium and beat in the remaining 1 cup sugar, then the vanilla. Blend in the eggs, one at a time, beating well after adding each. Beat in the cream just until it's completely blended. Be careful not to overmix!
Gently spoon the batter into the foil-wrapped springform and place it in a large shallow pan containing hot water that comes about 1 inch up the sides of the springform. Bake the cake at 350 degrees F until the edges are light golden brown and the top is slightly golden tan, about 1 1/4 hours. Remove the cake from the water bath, transfer to a wire rack, and cool in the pan for 2 hours, cover with plastic wrap, and refrigerate (while still in the pan) until it's completely cold, about 4 hours. Place in the freezer until ready to assemble the cake.
Meanwhile, prepare the devil's food cake layers. Check that the oven is preheated to 350 degrees F and that the water bath has been removed. Generously butter the bottom and sides of three 9-inch round layer cake pans. Very important: Line the bottom of all three pans with parchment or waxed paper (don't let the paper come up the sides).
Sift the flour, baking powder, and salt together in a small bowl. Cream the butter and both sugars together in a large bowl with the mixer on medium until light yellow and creamy. Add the egg yolks, one at a time, beating well after each. Beat in the melted chocolate and vanilla. Using a wooden spoon, stir in the flour mixture, alternately with the milk, mixing well after each until blended.
Put the egg whites and cream of tartar in a clean medium-size bowl and beat with clean, dry beaters on high until stiff (but not dry) peaks form. Fold about one-third of the whites into the chocolate batter until they disappear, then gently fold in the remaining whites. Don't worry if you still see a few white specks-they'll disappear during baking. Divide the batter evenly between the three pans. Bake until a toothpick inserted in the centers comes out with moist crumbs clinging to it, about 30 minutes. Cool the cakes in the pans on a rack for 15 minutes, then remove the cakes from the pans and gently peel off the paper liners. Let cool completely, about 2 hours, then cover with plastic wrap and refrigerate overnight or until ready to assemble the cake.
Early on the day you plan to serve the cake, make the chocolate curls and lay them out on the marble or baking sheet to dry. Now make the frosting: In a large bowl, sift the confectioners' sugar, cocoa, and salt together. In another large bowl, cream the butter with a mixer on high until light yellow and slightly thickened, about 3 minutes. With the mixer still running, beat in the chocolate, corn syrup, and vanilla. Reduce the mixer speed to low and beat in the sugar-cocoa mixture in two additions, beating well after each. Blend in the cream until the frosting is a spreading consistency, adding a little more cream if needed. Whip the frosting on high until light and creamy, about 2 minutes more.
To assemble the cake, remove the cheesecake from the freezer and let stand at room temperature about 10 minutes. Place one layer of devil's food cake, top side down, on a cake plate and spread with some of the frosting. Release and remove the ring of the springform, then remove the frozen cheesecake from the bottom of the pan. Place top side down on the frosted cake layer and spread with some frosting. Top with the second devil's food layer, top side down. Spread with more frosting and cover with the third devil's food layer, top side up. Brush away any crumbs from the sides and top of the fudge cake layers. Frost the sides and top of the cake with the remaining frosting, spreading the top with frosting about 1/2 inch deep. Use a long metal spatula that has been warmed under hot running water to smooth out the frosting on the sides and top of the cake. Coat the sides with the chocolate curls, placing them where you want. With a small spatula or table knife, swirl the frosting on top into a decorative design. Or, if you wish, decorate with a crosshatch design: Make about 6 vertical lines, 1 inch apart, then crisscross 6 more horizontal lines the same distance, swirling the icing a little as you go.
Refrigerate the cake for at least 2 hours to allow the cheesecake to thaw enough to easily slice. Use a sharp straight-edge knife, not a serrated one, to cut it. Cover any leftover cake with plastic wrap and refrigerate, or wrap and freeze for up to 1 month.