In the bowl of a stand mixer, add butter, both sugars, and mix until light and fluffy. Add eggs and vanilla extract. Mix until fully combined.
In a separate bowl, add flour, cream of tartar, baking soda, and salt. Whisk together dry ingredients well.
Fold dry ingredients together with the wet ingredients, until fully incorporated. Cover cookie dough tightly, and chill for at least 30 minutes or until dough is cold.
Meanwhile, mix the 4 TB sugar and 2-3 tsp cinnamon in a small bowl, whisk together, and set aside.
Preheat oven to 350F. Line baking sheets with parchment paper.
Form 1-inch balls of cookie dough, roll the balls in cinnamon-sugar coating, and place 2 inches apart on parchment-lined baking sheets.
Bake about 8 minutes or just until bottoms of cookies are golden brown. Sprinkle tops of hot cookies with a bit more cinnamon sugar. Let cool on baking sheet for about 5 minutes. Transfer to cool completely on wire rack.
Stir frosting in tub until smooth and silky. Spread desired amount of frosting onto the flat side of one cookie. Top with another cookie to create sandwiches. Repeat with remaining cookies.