Heat oven to 350°F. In large bowl, combine 1 cup of the sugar, shortening and butter; beat until light and fluffy. Add vanilla, almond extract and egg; blend well. Add flour, baking soda and cream of tartar; mix well.
Shape dough into 1-inch balls. Roll balls in remaining 1/3 cup sugar; place 2 inches apart on ungreased cookie sheets. With bottom of glass dipped in sugar, flatten balls to 1/8-inch thickness.
Bake at 350°F. for 7 to 12 minutes or until edges are light golden brown. Cool 1 minute; remove from cookie sheets.