In a medium mixing bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Add the 1 cup sugar, baking soda, and cream of tartar. Beat until combined, scraping sides of bowl occasionally. Beat in egg and vanilla. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour. Cover and chill in the refrigerator for 1 hour.
In a small mixing bowl combine the 2 tablespoons sugar and the cinnamon. Shape dough into 1-inch balls. Roll balls in the sugar-cinnamon mixture to coat. Place 2 inches apart on an ungreased cookie sheet.
Bake in a 375 degree F oven for 10 to 11 minutes or until edges are golden brown. Transfer cookies to a wire rack; cool. Makes about 36 cookies.