Place milk, 1 cup of cream, sugar, and salt in a medium sauce pan. Heat until the mixture is steaming, but not boiling. Remove from heat.
Add yolks to a medium bowl and very slowly whisk in warm milk. Transfer mixture back to saucepan and set over medium heat. Stir constantly until mixture thickens slightly and is 170 degrees. Remove from heat immediately so the mixture doesn't scramble.
Place remaining heavy cream in a large bowl and place strainer over bowl. Pour thickened yolk mixture through strainer to remove lumps. Stir with cream. Add vanilla extract. Cover with plastic wrap.
Chill mixture until completely cold, about 4-6 hours. Once chilled, ice cream base is ready for your machine. Freeze ice cream according to machine instructions. While the ice cream freezes, mix cream cheese, melted butter, powdered sugar, and vanilla extract together for the cream cheese frosting. Whisk to ensure there are no lumps. Add in red velvet cake pieces and mix to combine.
Once mixture is frozen, place half of ice cream in a storage container. Crumble half of the cake mixture over the ice cream. Repeat layer of ice cream and the rest of the cake pieces. Cover and freeze ice cream until solid, about 8 hours.