In a large bowl, stir together the Greek yogurt, cream, granulated sugar, and vanilla extract until the sugar is completely dissolved.
Cover the bowl and refrigerate for 1 hour to ensure the mixture is thoroughly chilled.
Pour the chilled mixture into an ice cream maker and freeze according to the manufacturer's instructions until it reaches a soft-serve consistency.
Transfer the frozen yogurt to a container, cover the surface with plastic wrap, and seal with a lid. Freeze for at least 2 hours or overnight to harden.