In a large bowl, use a stand mixer or a hand mixer and beat the softened butter and sugar on medium speed until combined and creamy, about 2 minutes. Add the egg and vanilla extract; mix on high until fully combined. Scrape down the sides and bottom of the bowl as needed using a rubber spatula. In a separate medium bowl, whisk together the flour, cream of tartar, baking soda, and salt. With the mixer running on low speed, gradually add the flour mixture to the butter mixture. Use a rubber spatula to scrape down the edges and bottom of the bowl, as needed. Once there are no more flour streaks, stop mixing. Do not over mix the dough. At this point, you can roll the cookie dough balls. However, the dough is a bit sticky and it is much easier to roll the dough when it is chilled. I also find the cookies bake up fluffier. Place the dough in the fridge to chill for 30 to 60 minutes.
Preheat oven to 400 degrees F. and prepare a few baking sheets with parchment paper, or line with a silicone baking mat. To roll the cookies: combine 1/3 cup sugar with the 1 1/2 tablespoons cinnamon in a small bowl. For each cookie, scoop out about 1 1/2 tablespoons of the dough and roll it into a ball. Roll the dough balls in the cinnamon/sugar mixture and place them 3 inches apart on a cookie sheet.
Bake cookies for 8 minutes and no more. The cookies may seem undercooked, but will continue to set after they are removed from the oven. Shape the cookies: Immediately after taking the cookies out of the oven, use a spoon or two (one in each hand) to push the edges of the cookies toward their centers. This makes the cookies round in shape and makes the center thicker and more chewy. You have to do this very quickly and immediately after taking them out of the oven, before the edges harden.
Allow cookies to cool on the pan for at least 3-5 minutes. While the cookies are still warm, but cool enough to handle (and not falling apart), place each cookie back into the bowl of cinnamon sugar. Coat one side of the cookie, then flip it over and coat the other side. This extra step is optional, but makes the cookies extra delicious! Allow the cookies to cool completely on a wire cooling rack. Enjoy! Cookies will stay fresh for up to 7 days, stored in an airtight container.