Preheat oven to 350 degrees. In a medium sized mixing bowl, combine flour, baking soda, cream of tartar, and salt. Set aside.
In a large mixing bowl, cream together butter, vanilla, sugar (hold back ¼ cup), powdered sugar, and milk. Add in eggs.
Slowly add in flour mixture until combined. The dough should look dry so if it’s still sticky add a bit more flour. Roll the dough into golf sized balls and place onto a non-stick cookie sheet.
To give the cookies their signature rough edge, dip the bottom of a glass into the reserved ¼ cup sugar and press onto the cookie ball to flatten the cookie. You still want the cookie to be thicker so don't press them to thin.
Bake for 10-15 minutes or until the edges are lightly brown.