Whisk the flour, cream of tartar, baking soda, and salt together in a bowl and set aside.
Beat the softened butter and 1 1/2 cups sugar with a mixer on medium for 3 minutes, until pale and fluffy.
Beat in the eggs one at a time, then the vanilla, scraping the bowl between additions.
Add the dry ingredients and mix on low just until no streaks of flour remain. Do not overmix.
Cover the bowl and refrigerate the dough for 30 minutes. Meanwhile, heat the oven to 375°F and line two baking sheets with parchment.
Stir the 1/4 cup sugar and cinnamon together in a shallow bowl. Roll the chilled dough into 1 1/2-inch balls and coat each one generously in the cinnamon sugar.
Arrange 12 balls per sheet, spaced 2 inches apart. Bake one sheet at a time for 9 to 10 minutes — the edges should be set and the centers should still look puffy and underdone.
Let the cookies cool on the hot pan for 5 minutes to finish setting, then move them to a wire rack.