Dilute cornstarch in 2 tablespoons of the root beer. Reduce remaining root beer in a saucepan over medium-high heat to 1 cup, about 30 minutes.Whisk cornstarch into reduction and cook until thickened, about 2-3 minutes. Remove from heat, cool completely, and chill.
Combine the milk, sweetener, and salt in a bowl. Stir to dissolve. Sprinkle gelatin across surface and allow mixture to stand until gelatin forms a light "skin," about 5 minutes. Place bowl containing milk mixture over a pot of simmering water and heat while stirring, until gelatin melts. Cool completely, but do not allow gelatin to set up.
In a separate bowl combine the yogurt and vanilla extract. Whisk together cooled milk mixture and the yogurt mixture. Cover and chill thoroughly, about 8 hours. Gelatin will set up during this time.
When ready to process, whisk yogurt mixture to break it up and remove lumps. Place in an ice cream machine and process according to manufacturers instructions. When yogurt is partially frozen, scrape into a chilled metal bowl, cover and freeze thoroughly, about 8 hours.
Serve frozen yogurt in chilled glasses or serving dishes, drizzled with sauce. Makes 8 serving.
Serving size: 3/4 cup frozen yogurt, 2 tablespoons sauce