In large bowl, combine butter, 1/2 cup sugar and 1/2 cup powdered sugar; beat well. Add egg, oil and lemon peel; mix well.
Lightly spoon flour into measuring cup; level off. Add flour, baking soda, cream of tartar and salt to sugar mixture; beat until well blended. Cover dough with plastic wrap; refrigerate 2 hours for easier handling.
Heat oven to 325. Shape dough into 1-inch balls. Place 2 inches apart on ungreased cookie sheets. Flatten cookies into 2-inch rounds with bottom of glass dipped in sugar.
Bake at 325 for 9 to 11 minutes or until edges begin to set and cookies are light golden brown. Cool 2 minutes. Remove from cookie sheets; place on wire racks. Cool 10 minutes or until completely cooled.
Meanwhile, in small bowl, blend 2 cups powdered sugar and enough lemon juice for desired spreading consistency. Spread frosting on cooled cookies. Sprinkle pistachios on frosting before it sets.