Cream butter and 1 1/2 cups sugar in a medium bowl. Add vanilla and eggs. Beat to combine thoroughly.
In a separate bowl combine flour, cream of tartar, baking soda and salt. Slowly add dry ingredients into wet ingredients, mixing until just combined.
Chill dough overnight.
Preheat oven to 350 degrees. Combine 1/3 cup sugar and chai powder in a small bowl. Roll 1 tablespoon of dough into balls (a cookie scoop is great for this), and roll them in the sugar/chai mixture. Place cookies 2 inches apart onto ungreased baking sheets.
Bake for 10-11 minutes. Transfer to a wire rack and let cool completely.