Add softened butter to the bowl of a stand mixer fitted with a paddle attachment. Whip butter on medium for 1 minute, until light and fluffy. Add sugars and mix together for another 2 minutes on medium. Add vanilla and egg and mix until combined.
In a separate bowl, whisk flour, cornstarch, baking soda, and salt together until combined. Add to butter mixture slowly, about 1/2 cup at a time. Once all dry ingredients are add, mix for 10 seconds longer until dough is completely formed. Add chocolate chips and mix for 5 seconds.
Place dough in a storage container to chill for at least 2 hours and no more than 3 days.
Once dough is chilled, remove from fridge. Preheat oven to 350 degrees. Line two baking sheets with parchment paper or silicone mats. Scoop dough into 1 1/2 tablespoons and form into balls. Push down to form into a uniform disk about 3/4 inch thick. Place on baking sheet about 2 inches apart. You shouldn't have more than 12 cookies per baking sheet.
Place cookies into a preheated oven and bake for 9-10 minutes until the edges are golden brown and still gooey in the center. Let them cool for 10 minutes and then transfer to cooling racks. Once the cookies are room temperature, place them in the freezer to chill for 10 minutes before assembly of the sandwiches and remove the ice cream from the freezer to soften a little. This will make it easier to handle.
Rip 12 squares of plastic wrap and layer them on the counter. Grab cookies from fridge. Place one cookie of a piece of plastic wrap. Scoop 1/2 cup of ice cream and place on cookie. Top with other cookie and gently push down. Immediately cover in plastic wrap tightly and then push cookies together a little more. Place in freezer to harden for an hour or so. Repeat with other cookies until all sandwiches are formed. Sandwiches are good in the freezer for 2 weeks.