Combine butter, brown and white sugar and mix until well combined. Beat in egg. Add vanilla and salt; mix.
In another bowl, add flour, baking soda and corn starch. Sift dry ingredients gradually into the wet ingredients. Stir until combined.
Heat barbecue to 350° degrees and grease two baking sheets.
Roll dough into 12 balls, about 2 tbsp of dough for each. Place in sheets about 2-3 inches apart.
Bake for 20 minutes or until cookies are barely turning golden brown (the softer the cookies the easier they will be to eat once frozen).
Add a couple of marshmallows to the top of each cookie as they cool. Allow to cool completely.
While the cookies are cooling, form the ice cream into cylinders by taking a measuring cup and lining it with plastic wrap. Fill the cup with ice cream. Wrap the plastic wrap completely around the ice cream, take it out of the cup and place in the freezer. Repeat until there is enough ice cream for half the cookies.
Once the cookies are finished cooling, decorate the cookies however you please (such as dipping them in chocolate, adding more marshmallows, sprinkling with graham cracker crumbs etc).
Place half of the cookies face down on a cookie sheet. Unwrap each ice cream cup and place on top of each cookie. Place the remaining cookies on top of the ice cream.
Enjoy immediately or wrap each sandwich in plastic wrap and store in the freezer for later!