1. In mixing bowl, cream the sugars and the butter. Mix in egg and set aside. Combine flour, salt, baking powder and cream of tartar. Add the oil and the flour mixture alternately to creamed sugars, mixing after each addition. Mix in vanilla. Cover and chill in refrigerator three to four hours or overnight. (If I am in a big hurry, which is most of the time, I just stick the dough in the freezer for about 30 minutes and then proceed with recipe.)
2. Roll dough into small balls, ¾ to 1 inch in diameter. Place on cookies sheet that has been lined with parchment paper and flatten with the bottom of a glass that has been dipped in sugar. (I have always had a problem in the past with the dough sticking to the bottom of the glass. I finally figured out a great trick - I place a small piece of parchment paper on top of the ball of dough, then flatten with the glass and there is no sticking - YEH!)
3. Bake at 350 for about 10-12 minutes or until golden brown. Cool on cooling rack. Store in an airtight container.
2 cups powdered sugar
¼ cup lemon juice
Whisk together the sugar and the lemon juice until smooth. Spread onto cookies to edges. Place cookies on cooling rack to allow excess icing to drip off.