Preheat oven to 350°F. Line baking sheets with parchment paper and set aside.
In a large bowl using a hand mixer or stand mixer fitted with the paddle attachment, cream together the butter and granulated sugar until light and fluffy.
Mix in the vanilla extract and egg until fully combined.
In a medium bowl, whisk together the flour, cocoa, cornstarch, baking powder, and salt.
Gradually add the dry ingredients to the butter mixture, mixing on low speed until fully incorporated. The dough should be soft but not sticky.
Roll out the dough on a lightly floured surface to about ¼-inch thickness.
Cut into desired shapes using cookie cutters. Transfer the cutouts to the prepared baking sheets, spacing them about 1 inch apart.
Bake for 8 to 11 minutes, depending on the size and thickness of the cookies. The edges should look set and the centers slightly soft; they will firm up as they cool.
Allow the cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Notes
I noted that I was able to get 34 cookies from this recipe. That will vary based on the thickness of your cookie dough and the size of your cookie cutter.