Chill the ice cream maker bowl for at least 24 hours before use.
In a large mixing bowl, whisk together heavy cream, whole milk, sugar, cocoa powder, red food coloring, vanilla extract, and kosher salt until smooth.
Pour the mixture into your ice cream maker and churn according to the manufacturer's instructions (about 20-25 minutes).
In the final minutes of churning, add the chopped red velvet cake pieces.
Prepare the cream cheese swirl by beating softened cream cheese until smooth, then gradually adding powdered sugar and mixing in vanilla extract. Refrigerate until ready to use.
Once the ice cream is churned, spoon half into a loaf pan. Add dollops of the cream cheese mixture and gently swirl with a spatula. Repeat with remaining ice cream and cream cheese mixture.
Cover with plastic wrap or wax paper pressed directly onto the surface and freeze for at least four hours until firm.
Let the ice cream sit at room temperature for a few minutes before scooping for the best serving consistency.
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Notes
Store in the coldest part of your freezer for up to two weeks, tightly covered to prevent ice crystals.