Preheat oven to 400ºF. If desired, line a cookie sheet with parchment paper for easy clean-up.
In a large mixing bowl or the bowl of a stand mixer, combine butter and sugar and use an electric or a stand mixer to beat for 2-3 minutes or until fluffy.
Add eggs and vanilla; beat until combined.
On top of the wet ingredients, add flour, cream of tartar, baking soda, and salt. Gently mix those together before mixing into the wet ingredients. Mix until combined, do not overmix. (See note regarding chilling dough.)
In a small bowl or plate, combine 3 tablespoons granulated sugar and 1 tablespoon cinnamon.
Use a 1 ½ tablespoon scoop to scoop out dough and roll into balls. Roll in cinnamon sugar mixture and place on cookie sheet. Place cookies about 2 inches apart on cookie sheet.
Bake for 10 minutes or until set. Be careful not to overbake. Let cool for 1-2 minutes on cookie sheet before removing to a cooling rack to cool completely.
Store baked cookies in an airtight container on the counter for up to a week, in the refrigerator for a couple of weeks, or in the freezer for 2-3 months.
To Chill or Not To Chill: I tested this recipe both ways. I found that chilled dough had a slightly increased bake time but did not change the appearance, taste, or texture of the cookies. If you prefer to use chilled dough, go for it, but I didn't find it necessary for this recipe.
I like to add ¼ - ½ teaspoon freshly grated nutmeg to the dough. It's not a classic snickerdoodle flavor but people seem to love it. It will give your cookies an edge over the ordinary.