In large bowl, beat butter, 1/2 cup granulated sugar and 1/2 cup powdered sugar with electric mixer on medium speed until light and fluffy. Beat in egg, oil and lemon peel until well blended. On low speed, beat in flour, baking soda, cream of tartar and salt until well blended. Cover with plastic wrap; refrigerate 2 hours for easier handling.
Heat oven to 325°F. Shape dough into 1-inch balls. On ungreased cookie sheets, place balls 2 inches apart. Dip bottom of small glass into 3 tablespoons sugar; press on cookies to make about 2 inches in diameter.
Bake 9 to 11 minutes or until edges begin to set and cookies are light golden brown. Cool 2 minutes; remove from cookie sheets to cooling racks. Cool completely, about 10 minutes.
Meanwhile, in small bowl, mix 2 cups powdered sugar and enough lemon juice until smooth and spreadable. Spread frosting on cookies. Sprinkle nuts on frosting before it sets.