In a bowl, with an electric mixer on medium speed, beat butter, oil, 1 cup granulated sugar, and the powdered sugar until very smooth. Beat in eggs until well blended, scraping down sides of bowl as needed.
In another bowl, mix flour, baking soda, cream of tartar, and salt. Stir or beat into butter mixture until well blended.
In a small bowl, mix remaining 1/2 cup granulated sugar and the cinnamon. Shape dough into 1-inch balls, then roll each in cinnamon-sugar mixture. Place balls 3 to 4 inches apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake in a 375° oven until edges of cookies are lightly browned, 12 to 15 minutes; if baking more than one pan at a time, switch pan positions halfway through baking.
Using a spatula, transfer cookies to racks to cool completely.
Note: Nutritional analysis is per cookie.