Preheat oven to 350°F. Line a baking sheet with parchment paper.
In a large bowl, beat butter, granulated sugar, and brown sugar until light and fluffy, about 2–3 minutes.
Add eggs one at a time, mixing well after each addition. Mix in vanilla extract.
Add flour, cornstarch, baking soda, salt, and ground cinnamon. Mix until just combined. Do not overmix.
In a small bowl, stir together the granulated sugar and cinnamon for the coating.
Scoop large dough balls (about ¼ cup each) and roll them in the cinnamon sugar mixture.
Place dough balls on the baking sheet, spaced apart. Bake for 9–11 minutes, until edges are set and centers are soft. Do not overbake.
Let cookies cool completely on the baking sheet.
In a bowl, beat cream cheese and butter until smooth. Add powdered sugar, vanilla extract, salt, and optional lemon juice. Beat until fluffy.
In a small bowl, mix the cinnamon sugar topping.
Frost cooled cookies, then lightly sprinkle cinnamon sugar over the frosting.