Preheat oven to 400 degrees F.
Cream together butter, shortening, 1 1/2 cups sugar, the eggs and the vanilla. Blend in the flour, cream of tartar, soda and salt. Shape dough by rounded spoonfuls into balls.
Mix the 3 tablespoons sugar and cinnamon. Roll balls of dough in mixture, then spoon cinnamon-sugar mixture over each cookie, turning once and doing the same on the other side. Place 2 inches apart on baking sheets lined with parchment paper.
Bake 8 minutes exactly! Remove from oven and let cookies cool on the baking sheets for an additional 2 minutes. Transfer to a cooling rack to cool completely.
Whisk all ingredients for vanilla swirl together in a small bowl. Let cool 5 minutes. Transfer to a small bag and snip the corner. Pipe glaze onto completely cooled cookies in the shape of a swirl.