Preheat oven to 350° and line two cookie sheets with parchment. Cream butter and sugars until light and fluffy, 3 to 4 minutes. Mix in egg and vanilla.
In another bowl, whisk together flour, cornstarch, baking soda, and salt, then add mixture to wet ingredients and mix until smooth. Dough will be thick.
Using a small cookie scoop, place tablespoon balls of dough onto parchment-lined cookie sheets. Press down lightly on each cookie to flatten slightly. Bake until edges are just starting to brown, 9 to 10 minutes.
Remove from oven and let cool 2 to 3 minutes, then transfer to a cooling rack to cool completely.
Make frosting: Beat cream cheese until smooth. Add powdered sugar and vanilla and mix until smooth.
Put sugar into a small bowl. Spread about a tablespoon of frosting onto tops of each cookie, then press into granulated sugar, coating frosting with sugar.
Just before serving, use a kitchen torch to caramelize sugar on top, then set aside to cool. Store cookies in the fridge for up to 4 days.