Preheat oven to 400°. In a bowl, beat butter and 3/4 cup sugar until blended. Beat in egg. In another bowl, whisk flour, cream of tartar, baking soda and salt; gradually beat into creamed mixture.
In a small bowl, mix brown sugar, cinnamon and remaining sugar until blended. Shape rounded tablespoons of dough into 1-in. balls; roll in brown sugar mixture. Place 2 in. apart on ungreased baking sheets.
Bake 7-9 minutes or until set and edges are lightly browned. Remove from pans to wire racks to cool completely.
Using a small scoop, place 2 tablespoons ice cream on bottom of a cookie; top with a second cookie, pressing gently to flatten ice cream. Wrap in plastic and freeze. Repeat with remaining ingredients. Freeze sandwiches overnight.