Preheat the oven to 350℉. Grease a 12-15 cup bundt cake pan with cooking spray and then dust with flour. Set aside.
In a large bowl, beat the butter and sugar with an electric hand mixer or stand mixer with fitted paddle attachment until light and fluffy, about 3-4 minutes, stopping to scrape the sides of the bowl. Add eggs, one at a time, beating well after each addition. Beat in vinegar and vanilla.
In a medium bowl, whisk together flour, cocoa powder, salt, and baking soda. Gradually add the flour mixture to the butter mixture and alternate with buttermilk, beginning and ending with the flour mixture, beating until just combined after each addition.
Stir in food coloring and spread about ½ of the mixture into the bottom of prepared pan.
For the filling: in another medium bowl, beat the cream cheese and sugar with a mixer at medium speed until smooth. Add egg and vanilla, beating until well combined. Spoon the filling on top of the batter in pan. Spread remaining batter onto the filling, making sure to cover it completely.
Bake until a toothpick inserted into the center of cake comes out mostly clean, about 60-75 minutes. Let cool in pan for 10 minutes, then remove from pan and let cool completely on a wire rack.