COOKIE DOUGH: Preheat oven to 350 degrees. Line baking sheets with parchment paper. In a large bowl, cream together shortening, butter and granulated sugar. Add pumpkin and eggs, mix well. In a separate bowl, sift together flour, baking soda, salt, ground cinnamon, pumpkin pie spice and nutmeg. Add pumpkin mixture to flour mixture and mix well. Dough will be moist.
Drop cookie dough by 1/4 cupfuls onto prepared baking sheet (6 cookies per sheet) flatten slightly and smooth the tops with dampened finger tips. Bake for 13-15 minutes. Cool cookies on the baking sheet over a rack. When completely cooled, drizzle with glaze using a fork or spreading with a knife. Top with sprinkles if desired.
ICING: Combine confectioners' sugar, melted butter, and vanilla. Add milk as needed, to achieve drizzling consistency.