Preheat oven to 350°F. Line cookie sheets with parchment paper.
Cream together butter, brown sugar, and 1/3 cup of sugar until fluffy. Add the egg and vanilla and mix until combined.
Add the baking soda, salt, and cornstarch and beat, then slowly add flour until just combined.
If dough is too soft to handle, chill for at least 15 minutes.
In a small bowl, combine 1/4 cup of sugar and 1 teaspoon cinnamon.
Using a small cookie sheet, roll cookies into balls and then roll in the cinnamon sugar mixture. Set on cookie sheet.
Bake for 10-13 minutes, until the edges are light golden brown. Cool completely while making frosting.
For the frosting, beat the butter with an electric mixer until smooth. Slowly add in powdered sugar until combined, then add salt, vanilla, and cinnamon. Add milk 1 tablespoon at a time until it's smooth and reaches your desired consistency.
Either pipe or spread frosting on cooled cookies and enjoy!