In medium bowl, mix shortening with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, brown sugar, pumpkin pie spice, baking powder and baking soda. Beat until combined, scraping sides of bowl. Beat in the egg and pumpkin puree. Beat in as much flour as you can with the mixer. Stir in remaining flour.
Drop dough by teaspoonfuls, 2 inches apart, onto ungreased cookie sheet. Bake at 375° for 8-10 minutes. Cool on wire rack and spread glaze.
For Brown Sugar Glaze, in small sauce pan, mix brown sugar, butter and milk. Heat and stir until butter melts. Stir in confectioner's sugar and vanilla. Spread over cooled cookies with wax paper underneath.