Preheat oven to 350°F (180°C). Butter and flour 2 (9-inch) round cake pans.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugar at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition.
In a medium bowl, whisk together flour, cocoa, baking soda, and salt. With mixer on low speed, gradually add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture, beating just until combined after each addition. Stir in food coloring, vinegar, and vanilla. Divide batter between prepared pans, smoothing tops with an offset spatula.
Bake until a wooden pick inserted in center comes out clean, 25 to 30 minutes. Let cool in pans for 10 minutes. Remove from pans, and let cool completely on wire racks.
Level cooled cake layers; reserve cake crumbs for garnish.
Remove sides of springform pan, leaving No-Bake Cheesecake on pan base. Place 1 cake layer, cut side down, on top of No-Bake Cheesecake. Top with a 12-inch cake board. Using cake board and base of springform pan, turn cake onto cake board. Remove pan base and parchment. Place remaining cake layer, cut side down, on top of No-Bake Cheesecake.
Reserve ¾ cup (161 grams) Eggnog Buttercream. Spread remaining Eggnog Buttercream on top and sides of cake. Place reserved buttercream in a pastry bag fitted with a large French star piping tip (Ateco #865). Pipe buttercream on top of cake as desired; garnish with cake crumbs, if desired.