In a medium saucepan over medium heat, combine heavy cream, milk, and granulated sugar. Stir continuously for about 5–7 minutes until the sugar completely dissolves and the mixture is warm but not boiling. Remove from heat and stir in vanilla extract. Allow to cool completely.

In a blender or food processor, blend fresh or frozen raspberries with a bit of sugar until smooth. Strain through a fine-mesh sieve to remove seeds. Set aside.

Transfer the cooled mixture to the refrigerator and chill for at least 1 hour.

Pour the chilled base into your ice cream maker and churn according to the manufacturer's instructions, typically for about 20–25 minutes until it thickens.

Divide the sherbet mixture into three bowls. Mix orange juice into one bowl, lime juice into another, and raspberry puree into the last. Stir each until well blended.

In a freezer-safe container, alternate spoonfuls of each flavored mixture and gently swirl together. Cover tightly and freeze for about 4–6 hours.

Scoop the sherbet into bowls and serve garnished with fresh fruit or mint.