Pre-heat oven to 400°F.
Line two large baking sheets with parchment paper.
In a large bowl, sift together the flour, cream of tartar, baking soda, and salt.
In a stand mixer, or with an electric hand mixer and a large bowl, cream the softened butter with 1½ cups sugar by beating them together for about 1½ minutes on medium, until fluffy and pale.
Add vanilla, and then the eggs, one at a time, and beat until well blended and smooth.
Gradually add in the flour mixture and continue beating until fully incorporated and the dough has come together, about 3 to 4 minutes.
In a medium-sized bowl, mix together the remaining ½ cup of sugar with the cinnamon.
Roll portions of the dough into generous 1 to 1½-inch balls with your hands (the dough will be soft). Roll each ball in the cinnamon-sugar mixture.
Place on the baking sheets, spaced about 2 inches apart.
Bake the cookies on the center two racks in the oven for 10 minutes. After about 5 minutes of baking, switch the sheets. The cookies will flatten on their own as they bake. If you made extra thick cookies, they may need to bake a couple more minutes. The cookies will harden somewhat as they cool.
Remove the baking sheets and allow to rest for 3 to 5 minutes. Transfer to baking racks and serve warm, or allow to come to room temperature.