Preheat oven to 375 degrees F. Line a cookie sheet with parchment paper and set aside.
In a medium bowl, whisk together flour, cornstarch, salt and cocoa powder. Set aside.
In a large mixing bowl using a stand or hand mixer, combine butter, oil, sugar and vanilla.
Add egg to butter mixture, mix on low. Small chunks of butter in the batter are okay. Scrape down sides.
With mixer on low, add dry ingredients to the wet ingredients. Mix until just combined, about 1 minute.
Roll out dough between two pieces of parchment paper at ¼-inch thickness. This techniques keeps the dough from sticking to the roller plus no need for a floured surface.
Cut dough into shapes with the cookie cutter of your choice.
Bake for 8-12 minutes. Cool completely on a rack before assembly.