Put the butter, sweetener, and half and half in a medium saucepan over medium-low heat. Cook for about 15 minutes, stirring occasionally until thickened and golden. Remove from the heat and whisk in the yogurt and salt. Cool to room temperature. Set aside ⅓ cup of the caramel sauce you just made.
Add 1 oz of the heavy cream to the remaining caramel sauce and mix with an electric mixer until smooth.
In a large bowl beat the remaining 7 oz of heavy cream until stiff peaks form. Gently fold the caramel and cream mixture into the whipped cream. Put into the container you will use to freeze it in. Pour or scatter bits of the remaining caramel over the top (mine was too thick to pour by this point). Cover and freeze for at least 6-8 hours.