In a large mixing bowl, beat the butter, oil, 1 cup granulated sugar, and powdered sugar until combined.
Beat in eggs and vanilla.
In a large bowl, whisk together flour, baking soda, cream of tartar, and 1 tsp. cinnamon.
In two additions, add flour mixture to the butter mixture, beating until just combined. Do not overmix.
Fold in cinnamon chips until evenly distributed.
Line baking sheets with parchment paper or silicone baking mats.
In a small bowl, whisk together remaining 1/2 cup granulated sugar and remaining 1 tsp. cinnamon.
Roll dough into 1-inch balls.
Thoroughly coat balls in the cinnamon-sugar mixture.
Arrange two inches apart on prepared baking sheets.
Bake at 375 for 8-11 minutes, or until edges and bottoms are lightly browned.
Remove to wire racks to cool.
Prepare icing.