1. Preheat oven to 375°F. With an electric mixer on high speed, beat butter and 3/4 cup sugar until light and fluffy, about 3 minutes. Add egg and vanilla and mix to combine. In a medium bowl, whisk flour, baking soda, cream of tartar and salt. With mixer on low, gradually beat in dry ingredients and continue mixing until well combined (dough will be thick).
2. Line 2 rimmed baking sheets with parchment paper. In a small bowl, combine remaining 3 Tbsp. sugar and cinnamon. Roll dough into 48 1/2-inch balls, then roll each ball in cinnamon sugar to coat. Arrange balls 1 inch apart on baking sheets and bake until puffed and light golden brown, about 10 minutes. Let cookies cool on baking sheets 5 minutes, then transfer to a wire rack and let cool completely. Repeat with remaining dough.
3. Spread 1 tsp. caramel sauce on bottom of 24 cookies. Lay cookies on counter, caramel side up. Top with 1 heaping Tbsp. ice cream, then second cookie, bottom side down. Wrap each ice cream sandwich in plastic wrap and place in freezer until firm, about 1 hour.