Heat oven to 375°. Grease large cookie sheet with shortening. Beat butter and brown sugar in large bowl with electric mixer on medium speed until smooth. Beat in egg, squash and vanilla until well blended (mixture may look curdled). Beat in flour, baking powder and 1 teaspoon pumpkin pie spice just until blended.
Drop dough by rounded teaspoonfuls about 1-inch apart onto cookie sheet. Bake 7 to 10 minutes or until centers are set. Cool 1 minute; remove from cookie sheet to wire racks. Cool completely, about 10 minutes.
Mix marshmallow creme and frosting in small bowl. Drop slightly less than 1/4 teaspoon of marshmallow mixture onto each cookie. Sprinkle with additional pumpkin pie spice.