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.