Place cream cheese in the bowl of a stand mixer fitted with the paddle attachment. Beat on medium for 1 minute until smooth.

Scrape down the sides of the bowl, add sweetened condensed milk, vanilla, and salt and continue beating until combined and smooth. Scrape this mixture to a large bowl.

Add cream to the stand mixer bowl (no need to clean it out) and using the whisk attachment, whip the cream to stiff peaks, about 3 minutes.

Add a third of the whipped cream to the cream cheese mixture and fold to combine. Add remaining cream and gently fold until completely incorporated. Scrape into a 9x13-inch baking pan and smooth top. Cover and freeze for 2 hours.

Drop tablespoons of sauce about 1-inch apart on top of the ice cream base and sprinkle churro pieces over the sauce. Use a skewer to create a swirl pattern and mix in churros.

Cover and return to the freezer until solid, at least 2 hours more. Once solid, transfer to a container with a tight lid.