Make the New York Cheesecake Swirl:
In the bowl of an electric mixer fitted with a paddle attachment, beat the sugar, cream cheese, and vanilla extract on high speed until well combined. Add the eggs and continue to beat on high speed. Make sure to scrape down the sides to ensure incorporation of all the ingredients. Once combined, gently fold in the chocolate morsels. Chill the mixture for approximately 1 hour.
Make the Rich Chocolate Cake:
Preheat the oven to 350 degrees F.
Lightly grease and flour 3 (9-inch) cake pans and set aside. In a large bowl, sift the flour, cocoa powder, baking soda, and baking powder together. Add the sugar and salt and stir to combine well.
In a medium bowl, whisk together the eggs, buttermilk, vanilla and melted butter. Add the wet ingredients to the dry ingredients in small increments, gently folding the mixture until well combined.
Pour the mixture into 1 of the prepared cake pans, only filling the pan halfway. Randomly place teaspoon-size dollops of the cream cheese mixture atop the chocolate cake batter. Repeat with the remaining 2 prepared cake pans. Bake for approximately 30 minutes, or until a wooden toothpick inserted in the center comes out clean. Cool completely on a wire rack.