Place the mango cubes in your food processor’s bowl and process until completely smooth, stopping once to scrape the sides and bottom of the bowl.
Add the yogurt, sweetener and vanilla and process until combined, stopping once or twice to scrape the sides and bottom of the food processor bowl. The mixture will start out grainy, especially if you use frozen mango. Keep processing, and it will eventually become nice and smooth.
Process for 1 more minute, until light and fluffy.
Using a rubber spatula, transfer the mixture into a square 9-inch pan. Cover with cling wrap, and freeze for 45 minutes.
Remove the pan from the freezer. Use a rubber spatula to stir the already-frozen edges into the still-soft center, then mix everything together until smooth.
Cover again, and place in the freezer for 2 more hours, repeating the process of stirring the yogurt every 30 minutes.
When the entire mixture is frozen, serve immediately as a soft-serve yogurt (use chilled bowls), or freeze for 1-2 more hours for a firmer consistency that you can scoop out with an ice cream scoop.
If you end up storing the frozen yogurt for longer than that (say, overnight or a couple of days), remove it from the freezer 20 minutes before serving, to allow it to soften.
Video
Notes
You can use frozen mango chunks instead of fresh. In the video, I used a 10-oz bag of frozen mango chunks. Keep in mind that the food processor will have a very hard time processing them, so hold it tightly while it works - its body as well as its lid.Another option is to partially thaw the mango chunks in the microwave prior to processing them.You can replace the stevia with a powdered sweetener or with ⅓ cup honey.