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As the oven is heating up, line your baking sheet with parchment paper. If you don't have any parchment paper, non-stick cooking spray will do in a pinch, or a little melted butter. You want it to be soft, but not liquidy. Room temperature should be fine, as it will be easy to blend. If you need, put the butter briefly in the microwave — 4-5 seconds, tops. Mix everything well so that there are no clumps and everything is well distributed. Cream of tartar leads to a light tang and helps the cookies get fluffy. However, if you want crispier cookies or don't want this tangy edge, you can skip the cream of tartar. Mix the butter/margarine and sugar together until fluffy. Keep your electric beaters on medium speed and work the softened butter until it is a nice whipped, semi-stiff mixture. All of the sugar should be incorporated into the butter. This should only take 3-5 minutes of mixing. Keep the mixer going until the eggs are completely incorporated into the butter mixture. Use a rubber spatula to scrape the sides of the bowl as necessary, making sure everything is well mixed. Want tangier, less sweet cookies? Skip the vanilla. Add the flour in small chunks, keeping the mixer going the entire time. This prevents flour "explosions," where the beaters scatter the flour instead of mixing it. As you add flour, keep scraping the sides of the bowl to make sure everything is getting added in. After dough is thoroughly mixed, chill it for 30-45 minutes to guarantee the best results. Start with 2 tablespoons of each, adding more as necessary. Feel free to play with this ratio as well -- while few chefs use more cinnamon than sugar, many will add a few extra parts sugar to avoid overly spiced cookies. You can use a small spoon to measure out the dough, then roll it between your palms to form the dough balls. You then roll the dough in the cinnamon sugar to coat it completely. Make sure the cookies are at least 2 inches apart, as they will flatten and spread as they cook. A good snickerdoodle has a light, chewy texture, not a crunch. You want to try and keep the tops of the cookies light, so remove them when the outsides are just getting golden for the best cookies. Let them cool on the baking sheet for 2-3 minute before eating.

Summary:
Preheat the oven to 400 °F (205 °C). Let the butter soften outside the fridge for 5-10 minutes. Whisk together the flour, cream of tartar, baking soda, and salt in a large bowl. Using an electric mixer, cream the butter in with the sugar until completely mixed. Add the vanilla extract and eggs and beat for 2-3 minutes. Turning your mixer to "low" speed, slowly add the flour mixture. In a shallow bowl, mix equal parts white sugar and ground cinnamon. Roll the chilled dough into 1" wide balls, then roll each ball in cinnamon sugar. Set the cookies on the parchment-lined baking sheets and cook for 8-9 minutes. Remove from the oven when the outside edges are just turning golden.