Place the cream cheese, heavy cream, milk, sugar, food coloring, cocoa powder, and salt in a high powered blender. Pulse until fully combined and smooth. Pour into a large bowl and chill for a minimum of 2 hours or up to overnight in the refrigerator.
8 ounces cream cheese, 1 cup heavy cream, 1 cup whole milk, ¾ cups granulated sugar, 4 teaspoons red food coloring, 1 tablespoon unsweetened cocoa powder, ¼ teaspoon kosher salt
Once chilled, pour the ice cream mixture into an ice cream maker and process according to manufacturer's instructions.
While the ice cream is churning, prepare the icing: In the bowl of a stand mixer fitted with the paddle attachment, cream the cream cheese, butter, and vanilla together. Add in the powdered sugar, ½ cup at a time, until fully combined and fluffy. Set aside.
½ cup unsalted butter, 4 ounces cream cheese, 1 teaspoon pure vanilla extract, 2 cups powdered sugar
When ice cream has processed, pour half of it into a freezer-safe container and top with 5-6 spoonfuls of the icing. Use a knife to loosely swirl. Top with the rest of the ice cream and then more icing, swirling once more.
Allow to freeze overnight before serving.