Preheat oven to 350 degrees. Line several baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer, cream together the shortening, butter and oil until smooth. Add sugars and beat for 2 minutes, until light and fluffy. Add eggs, incorporating one at a time. Mix in vanilla.
In a separate bowl, combine the flour, cornstarch, baking powder and kosher salt. Slowly add to the creamed mixture until just combined, scraping the sides of the bowl when needed.
Mix in the remaining ingredients by hand. Scoop onto prepared baking sheets using a standard cookie scoop. Bake for 9 to 11 minutes, until just set and barely golden on the edges. Allow to cool for 2 minutes on the pan then remove to a rack to cool completely.