Cream together shortening and 1 cup sugar. Note: I have recently experimented by substituting 1/4 cup of the shortening for a healthy, butter-flavored margarine -- using all butter or margarine tends to make the cookies spread out and not hold their shape, however). Add pumpkin and egg and cream lightly until fluffy (I usually use a hand mixer for this).
Combine flour, soda, cinnamon and salt; mix into cream mixture. (I usually switch to a wooden spoon, to keep the mixture light and fluffy -- you don't want to overbeat).
Fold in butterscotch chips.
Drop by teaspoons on an ungreased cookie sheet, and bake at 375 degrees for 10 to 12 minutes or until done (I sometimes find it takes a little longer, depending on humidity, the size of the cookie, etc.). Refrigerate remaining dough, covered, while other cookies are baking.
In the meantime, make the frosting: combine the butter and milk in a small saucepan and cook until sugar is dissolved (or, you can cook in the microwave, about 30 seconds+/-). Let cool for several minutes (mixture will be still be warm). Stir in the powdered sugar, sifted, then add vanilla; whisk well until smooth.
Frost cookies fresh out of the oven and while frosting is still somewhat warm; frosting will harden as it sits out; if necessary, reheat for a few seconds in the microwave until pliable again and/or add a few more drops of milk and restir.