In a large bowl using a handheld mixer or stand mixer with paddle attachment, beat the butter, brown sugar, and granulated sugar together on medium speed until creamy, about 2 minutes. Beat in the egg and vanilla extract. Scrape down the sides and bottom of the bowl as needed.
In a separate bowl, combine flour, cornstarch, baking soda and salt. Pour dry ingredients into the wet ingredients, then beat on low speed until combined. The cookie dough will be slightly thick. On low speed, beat in the chocolate chips. Cover dough tightly with aluminum foil or plastic wrap and chill for at least 1 hour and up to 2 days. Chilling is mandatory for this cookie dough.
Remove cookie dough from the refrigerator and allow to sit at room temperature for 10 minutes. Preheat oven to 350°F (177°C). Line 2 large baking sheets with parchment paper or silicone baking mats. Set aside.
Once chilled, the dough will be slightly crumbly, but will come together when you work the dough with your hands. Roll balls of dough, about 1.5 Tablespoons of dough each, into balls. (I like to use this medium cookie scoop.)
Bake for 10-11 minutes, until barely golden brown around the edges. The cookies will look extremely soft when you remove them from the oven.
Turn the oven up to high broil. (If you know your oven gets overly hot on high, turn on a low broil. I usually broil on high.)
Top each cookie with a marshmallow half, sticky side of the marshmallow down on the cookie. Return to the oven for only 1-2 minutes or until the broil setting lightly toasts the marshmallow. Keep a very close eye on it—it toasts FAST!
Remove cookies from the oven and cool for 5 minutes on the cookie sheet as you prepare the chocolate. Using the back of a spoon, press the marshmallow down if it didn’t melt down on its own. (Sometimes it holds its taller shape, but gently pressing down flattens it out.)
Melt the chocolate You can melt the chocolate in a double boiler or the microwave. If using the microwave: place the chopped chocolate in a medium heat-proof bowl. Melt in 20 second increments in the microwave, stirring after each increment until completely melted and smooth. Spoon evenly over marshmallow. Top with graham cracker crumbs. Allow chocolate to completely set at room temperature, about 60 minutes, or in the refrigerator, about 30 minutes.
Cookies stay fresh covered at room temperature for up to 3 days or in the refrigerator for up to 1 week.
