Preheat oven to 350° and line two cookie sheets with parchment paper.
Cream butter and sugars for 3 minutes. Mix in egg and vanilla.
In another bowl, whisk together flour, cornstarch, baking soda, and salt. Add mixture to wet ingredients and mix until smooth.
Using a small cookie scoop, place balls of dough onto cookie sheets and press down to flatten slightly. Bake until edges just start to brown, 9 to 10 minutes.
Remove from oven and let cool for 5 minutes.
Turn oven to BROIL and make frosting: Beat cream cheese until smooth. Add powdered sugar and vanilla and mix until smooth.
Spread about a TSP of frosting onto tops of each cookie, then sprinkle with sugar.
Place cookies back in oven and watch the closely-- roughly 1 minute-as sugar caramelizes on top.
Set aside to cool and enjoy!