Position a rack in the center of the oven and preheat to 350°F (177°C). Line two baking sheets with parchment paper or silicone mats.
In the bowl of a stand mixer fitted with the paddle (or using a large bowl and hand mixer), beat the softened butter with the brown sugar and granulated sugar on medium speed until the mixture is pale and fluffy, about 3 to 4 minutes.
Add the egg and vanilla to the butter-sugar mixture and mix until smooth, scraping down the sides of the bowl so everything is evenly combined.
Whisk together the cocoa powder, flour, baking soda, salt, cornstarch, and instant espresso powder in a separate bowl until uniform.
With the mixer on low, add the dry ingredients to the wet in two additions, mixing just until the dough comes together and no streaks of flour remain. Avoid overmixing to keep the cookies tender.
Fold in the drained maraschino cherries, dried cherries, and chocolate chips using a spatula so the mix-ins are evenly distributed without crushing the cherries.
For chewier, taller cookies, cover the bowl and chill the dough in the refrigerator for about 20 minutes. Chilling firms the dough and helps the cookies spread less while baking.
Scoop rounded 2-tablespoon portions of dough (a medium cookie scoop) and place them 2 to 3 inches apart on the prepared sheets. Keep the dough mounded rather than flattened. Press a few extra chocolate chips onto the top of each mound if you like.
Bake one sheet at a time in the center of the oven for 9 to 12 minutes, until the edges are set and the centers still look soft and slightly underbaked. Rotate the sheet halfway through if your oven has hot spots.
Allow the cookies to cool on the baking sheet for 5 minutes so they finish setting, then transfer to a wire rack to cool completely. The centers will firm up as they cool, yielding a chewy texture.