Freeze the bowl of your ice cream maker 24 hours in advance.
Whisk together all the ingredients: Combine the yogurt, heavy cream, and sweetener in a bowl. Whisk until heavy cream and the sweetener are fully mixed in, 1 to 2 minutes. If you’re using granulated sugar, then the mixture should feel smooth between your fingers, not gritty.
Refrigerate, if needed: If your yogurt mixture has warmed at all or if you want to wait to churn it for any reason, cover it and place it in the fridge until it has cooled to fridge temperature again. If your ingredients have only been out of the fridge a few minutes, another chilling isn’t necessary.
Churn the yogurt until thick: Pour the yogurt base into the ice cream machine. Churn until the yogurt has thickened to the texture of soft-serve or a thick milk shake (17 to 20 minutes on most machines).
Scoop into a freezer container: Smooth the top. Press a piece of parchment against the top of the yogurt to prevent ice crystals from forming.
Freeze until hard, at least 6 hours or overnight.
Serve the frozen yogurt: When ready to serve, let the frozen yogurt sit on the counter for 20 to 30 minutes, until just soft enough to scoop. Use long, shallow strokes to form balls of frozen yogurt. Serve right away.