Preheat oven to 350 degrees F. Thoroughly grease and flour a 12-cup bundt cake pan. Set aside.
Start with the cheesecake filling. With an electric mixer on medium speed, cream together the cream cheese, powdered sugar, egg, and vanilla in a bowl. Set aside.
In a mixing bowl, whisk together the flour, unsweetened cocoa powder, baking soda, baking powder, and salt.
In a large mixing bowl, using a hand mixer on medium speed, cream together the butter and sugar until they are well combined, 2 to 3 minutes.
Add the buttermilk, canola oil, eggs, vanilla, vinegar, and red food coloring. Mix on medium-low speed until everything is well combined.
Add the dry ingredients to the wet and mix on medium-low speed just until everything is mixed together. Avoid overmixing.
Pour 1/2 of the red velvet cake batter into the prepared bundt pan. Then, spoon the cheesecake filling onto the center of the cake batter keeping the filling from touching the edges (see photo in the post).
Pour in the rest of the cake batter covering the top and sides of the cheesecake filling. Give the cake a gentle tap on the counter to help everything settle and remove any air pockets.
Bake the cake for 50 to 60 minutes. Start checking at 50 minutes and remove once there are still some moist crumbs on the stick. Cool the cake in the pan on a wire rack for 2 hours. Then, remove the cake and cool for another hour.
In a bowl, with a hand mixer on medium speed, cream together the cream cheese and powdered sugar for the frosting until they are light and airy, 2 to 3 minutes. Add the milk and mix on medium speed for another minute or until the frosting is smooth.
Pour or spoon the frosting over the cooled bundt cake pushing it down the sides to create drips.
Notes
I prefer to use softened butter and a pastry brush to grease my bundt pan before coating it in flour. You can also use oil or cooking spray. Just make sure you get the grease everywhere and flour the pan well.
Depending on your oven, it can take anywhere from 50 to 60 minutes to fully bake. Begin checking at 50 minutes and watch it closely towards the end to avoid overbaking.
Allow the cake to cool in the pan for the first 2 hours. It comes out much easier and cleaner that way.
Store the cake in the fridge for up to 5 days. Allow it to come back to room temperature before serving for the best results.