Cream shortening in a medium mixing bowl; gradually add sugar, beating until light and fluffy. Add beaten egg, beating well. Combine flour, soda, and salt in a small mixing bowl; add to creamed mixture, stirring well. Stir in mincemeat and vanilla.
Drop dough by teaspoonfuls 2 inches apart onto greased cookie sheets. Bake at 375° for 10 to 12 minutes. Cool slightly on cookie sheets. Remove to wire racks to cool completely.