Cream the butter, oil, sugar, vanilla, and salt in a bowl with a mixer until the mixture is smooth. (Do not overbeat, as this will incorporate air into the dough, which may lead to spreading.)
Add the egg and mix just until incorporated.
Mix in the cornstarch, cocoa powder, and 1 1/4 cups of the flour.
Continue adding more flour until the dough gathers itself into a ball and pulls away cleanly from the sides of the bowl. (You may not need all the flour.)
Preheat the oven to 375 degrees F.
Roll the dough to a thickness of 3/8-inch with a rolling pin, between two sheets of parchment paper.
Cut shapes using a round cookie cutter. (I used a cookie cutter in 2-inch diameter)
Bake on parchment-lined baking sheets for 9 to 12 minutes, or until the cookies look dry on the surface, and feel firm around the edges, and set in the middle.
Cool completely.