Combine the cream, milk, brown sugar, salt, vanilla bean paste, and vanilla extract in a mixing bowl. Whisk to combine.
Pour into the ice cream maker. Churn according to the manufacturer's instructions. Transfer the finished ice cream to an airtight container and place it in the freezer until ready to serve.
Notes
If you don't happen to have dark brown sugar on hand, you can add 1 teaspoon of molasses to the mixture to create the same richer flavor.