Preheat oven to 350°F. Line two baking sheets with parchment paper.
Using a hand mixer, or stand mixer fitted with the paddle attachment, beat butter, brown sugar, and granulated sugar for several minutes, until light and fluffy. Add the the egg and vanilla extract and beat to combine, scraping the sides of the bowl as necessary.
In a separate bowl, combine the cocoa, flour, baking soda, salt, and cornstarch. With the mixer on low speed, add the flour mixture to the butter mixture, mixing until combined. Stir in maraschino cherries, dried cherries, and chocolate chips.
Scoop the dough into 2 tablespoon portions and place a couple inches apart on prepared cookie sheets**. Make sure the dough balls are mounded fairly high (not flattened). If desired, press a couple chocolate chips onto the tops of each dough ball. Bake for 8-11 minutes, until the edges are just beginning to set and the center is still underdone. They will look under baked and too soft to pick up, but the cookies will set as they cool. Place the baking sheets on wire racks to cool completely.