Prepare the Cheesecake Layers: Preheat oven to 325°F. Line bottom and sides of 2 (9-inch) round pans with aluminum foil, allowing 2 to 3 inches to extend over sides; lightly grease foil. Beat cream cheese and granulated sugar with an electric mixer on medium speed just until completely combined, about 1 minute. Add eggs, sour cream, buttermilk, cocoa, red food coloring, vanilla, and vinegar, beating on low speed just until fully combined, about 4 minutes. (Do not overbeat.) Pour batter into prepared pans.
Bake at 325°F for 10 minutes; reduce heat to 300°F, and bake until center is slightly jiggly, about 1 hour and 5 minutes. Turn oven off. Let cheesecake layers stand in oven 30 minutes. Remove cheesecake layers from oven; cool in pans on wire racks 1 hour. Cover and chill 8 hours.
Prepare Vanilla Cake Layer: Preheat oven to 325°F. Beat butter with a heavy-duty stand mixer on medium speed until creamy; gradually add granulated sugar, beating until light and fluffy, about 5 minutes.
Stir together flour, baking powder, and salt in a bowl; add to butter mixture alternately with milk, beginning and ending with flour mixture. Beat on low speed just until blended after each addition. Stir in vanilla.
Beat egg whites in a clean bowl with an electric mixer fitted with whisk attachment on high speed until stiff peaks form; fold about one-third of egg whites into batter. Gradually fold in remaining egg whites. Pour batter into 1 greased (with shortening) and floured 9-inch round cake pan.
Bake at 325°F until a wooden pick inserted in center comes out clean, 33 to 36 minutes. Cool in pan on a wire rack 10 minutes. Remove from pan to wire rack; cool completely, about 30 minutes.
Prepare the Cream Cheese Frosting: Beat cream cheese and butter with an electric mixer on medium speed until creamy, about 5 minutes. Gradually add powdered sugar, beating at low speed until blended after each addition; stir in vanilla. Increase speed to medium, and beat until light and fluffy.
Assemble Cake: Lift Cheesecake Layers from pans, using foil sides as handles. Place 1 Cheesecake Layer, bottom side up, on a cake plate. Place Vanilla Cake Layer on Cheesecake Layer. (This layer will be slightly taller than the Cheesecake Layers; trim the top of the layer, if desired). Top Vanilla Cake Layer with the remaining Cheesecake Layer, bottom side up. Spread a thin layer of Cream Cheese Frosting over top and sides of cake to seal in crumbs. Chill cake 20 minutes. Spread remaining frosting over top and sides of cake.
Garnish How-To: Curl the chocolate: Slightly soften an 8-oz. bar of white chocolate by heating it in the microwave at 50% power for 10 seconds. Shave the chocolate into large curls using a vegetable peeler. Pile the curls on top of the frosted cake.