In a medium saucepan, combine heavy cream, milk, and granulated sugar over medium heat. Stir continuously for about 5-7 minutes until the sugar completely dissolves. Remove from heat and stir in vanilla extract. Allow to cool completely before refrigerating to chill.

While the base is chilling, prepare the raspberry puree. Blend raspberries with a bit of sugar to taste until smooth and strain the mixture to remove seeds.

Once the creamy base is chilled, pour it into your ice cream maker and churn for 20-25 minutes until it reaches a soft-serve consistency.

Divide the churned mixture into three bowls. In the first bowl, mix in orange juice; in the second, add lime juice; and in the third, fold in raspberry puree.

In a freezer-safe container, alternate scoops of each flavor to create a layered effect. Swirl gently, cover, and freeze for 4-6 hours until firm.

To serve, let sherbet sit for a few minutes to soften. Use an ice cream scoop to serve into bowls or cones.