Preheat oven to 350°F. Combine 2 Tbsp sugar with ½ tsp cinnamon for rolling, and set aside.
Cream butter and sugar until light and fluffy, about 2 minutes with a mixer.
Add egg and vanilla, and mix until thoroughly combined.
Combine flour, cornstarch, cream of tartar, baking soda, baking powder, cinnamon, and salt.
Add flour mixture to egg and butter mixture in three parts, mixing thoroughly between additions.
Once dough is just combined, divide into 6 balls, rolling each ball in cinnamon sugar mixture.
Arrange on a baking sheet with plenty of space between cookies, and press down on dough balls lightly with the bottom of a glass until they resemble hockey pucks.
Bake for 9 minutes, or until edges have just begun to color.
Let cool on a baking rack.
Beat softened cream cheese until smooth and soft.
Add butter, and beat until thoroughly combined.
Add vanilla and confectioners sugar and pulse until combined.
Add milk as needed until desired piping consistency is reached.
Frost cookies, then sprinkle with cinnamon sugar.