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 and sugars 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, slowly add the flour mixture to the butter mixture, mixing until just combined. Reserve ¼ cup chocolate chips for topping the dough balls, stir in the remaining ¾ cup chocolate chips.
Scoop the dough into 1-½ tablespoon portions (I use a cookie scoop) and place onto prepared cookie sheets. Make sure the dough balls are mounded fairly high (not flattened). Press some of the reserved chocolate chips onto the tops of each dough ball.
Bake for 7-8 minutes**, the cookies will just be beginning to spread and still have a mound in the center. Remove from the oven and carefully press one half of a marshmallow onto the top of each cookie, pressing it slightly to flatten out the cookies just a bit. Return to oven for another 1-2 minutes. The edges should appear almost done, but still a bit under baked - that's ok. Do not bake them until they look completely done, they should appear under baked when you pull them from the oven. Place the baking sheets on wire racks to cool completely (the cookies will firm up as they cool). If desired, sprinkle a little chopped chocolate on top of each cookie while they are still warm. Repeat with remaining dough once the baking sheets have cooled***.