Cream the butter and sugar. In a stand mixer with the paddle attachment (or a large bowl with a hand mixer), beat the softened butter and 1½ cups sugar together until light and fluffy. Scrape down the sides of the bowl — the butter likes to stick to the edges — and mix again briefly.
Add the eggs and vanilla. Crack in both eggs and add the vanilla. Mix until well combined, then scrape down the sides again and give it one more quick mix to make sure everything is incorporated.
Combine the dry ingredients. In a separate bowl, stir together the flour, baking soda, cream of tartar, and salt. You don't need to sift — just give it a quick stir so the baking soda and cream of tartar are spread throughout the flour.
Add dry ingredients to wet. With the mixer on low speed (so your flour doesn't puff out everywhere), gradually add the dry ingredients. Mix until just combined — don't overmix. You should have a soft, slightly sticky cookie dough.
Make the cinnamon sugar coating. In a small bowl, stir together ⅓ cup sugar and 2 tablespoons cinnamon. If you have a very strong cinnamon, go just a little lighter on the measurement.
Scoop and roll. Using a cookie scoop or two teaspoons, scoop the dough into balls. Roll each ball in the cinnamon sugar mixture — I like to roll mine through twice to get a really good coating. Place on a parchment-lined or ungreased cookie sheet, leaving about 2 inches between cookies.
Chill (optional but recommended). For thicker, puffier cookies, place your cookie sheet in the refrigerator for 30 minutes to 1 hour. If you're in a hurry or prefer a thinner cookie, you can skip this step and bake right away.
Bake. Preheat your oven to 350°F. Bake for 8 to 10 minutes. At 8 minutes you'll get a softer, chewier cookie — they may look slightly underdone, but they'll set up as they cool. At 10 minutes you'll get a crispier cookie. Let them cool on the baking sheet for a few minutes before transferring to a wire rack.