To make cake: Preheat oven to 350ºF. Prepare 3 6-inch cake pans by greasing with butter, adding parchment paper circles to the bottom. Butter the papers and dust with flour, tapping out any excess.
In a large bowl, whisk together eggs, oil, buttermilk, vinegar, and vanilla. Then stir in food coloring.
In a separate bowl, whisk together flour, sugar, cocoa powder, salt, and baking soda. Slowly whisk in the dry ingredients to the egg mixture until completely combined.
Divide batter evenly between cake pans and then drop the pans on the counter a few times to release any air bubbles. Bake for about 30 minutes, or until a toothpick inserted comes out clean. After about 10 minutes, remove from pans and cool completely on a wire rack.
To make icing: In a large bowl using a mixer, beat cream cheese and butter on medium speed. Add melted chocolate and vanilla, and mix well. Add sugar gradually, beating until fluffy.
Garnish with white chocolate shavings and crumbled green velvet cake.