Preheat oven to 325F and spray a bundt pan thoroughly with baking spray. Alternatively, you can butter the pan and dust it with flour. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large mixing bowl with an electric hand mixer), beat together butter and cream cheese until smooth.
Add sugar to the bowl, mixing until well-combined.
Add eggs, one at a time, mixing after each addition. Scrape down the sides of the bowl as needed.
Add vanilla extract, mixing just until incorporated.
With the mixer on low speed, add cake mix and flour, mixing just until combined.
Add in the water and red velvet emulsion or food coloring (if using), mixing to incorporate well.
Pour the batter into the prepared bundt pan, smoothing it out evenly. Bake in preheated oven for 60-65 minutes, or until a toothpick inserted in the center of the cake comes out clean.
Remove from oven and let cool in the pan for about 10 minutes. Then, run a knife along the edges of the cake to ensure it separates from the pan. Invert cake onto a serving plate, then allow to cool completely before icing.
Once the cake is cooled, prepare the icing by combining cream cheese, butter, powdered sugar, and vanilla extract in a medium mixing bowl. Beat with an electric mixer until smooth.
Add four tablespoons of milk, mixing well to combine. If necessary, continue adding one or two more tablespoons of milk until desired consistency is achieved. You want the icing to be thin enough to spoon over the cake, but not so thin that it runs completely off the cake.
Spoon the icing over the cake, allowing it to drip partly down the sides.
Serve and enjoy!