Heat oven to 400 degrees F. Place a cooling rack on the countertop for cooling the cookies.
Beat butter on medium speed with an electric mixer. Add sugar and beat again until well combined.
Crack the eggs in a separate bowl and make sure no shells make it into the dough. Pour the eggs into the mixing bowl with the butter and sugar. Add the vanilla. Beat on medium speed until mixture is creamy. You may need to scrape the sides of the bowl if the mixture has pasted onto the sides.
In a separate mixing bowl, whisk together flour, cream of tartar, baking soda, and salt. Add gradually to creamed mixture at low speed. Mix just until blended.
Shape dough into 1-inch balls.
Combine 1/2 cup sugar with 2 tablespoons cinnamon in a small bowl. Roll balls of dough in the cinnamon-sugar mixture. Place 2 inches apart on an ungreased baking sheet.
Bake for 7 to 8 minutes. Do not overbake. Cool for 2 minutes on the baking sheet. Remove cookies to cooling racks to finish cooling.