Measure flour, cream of tartar, and baking soda together and set aside. Beat vegetable shortening and sugars together until fluffy. Add vanilla, salt, and egg to sugars and butter mixture beat until smooth.
Add flour, baking soda, and cream of tartar to sugar and shortening mixture about a third at a time. Beat until completely mixed through. The dough will seem a little thick.
Stir pecans into cookie dough with a spoon and blend well. Chill dough for 30 - 45 minutes. Roll cookie dough into a 1 - 1 1/4 inch balls and place on an ungreased cookie sheet. Flatten balls to 1/4 inch thick. Bake in a preheated oven for 12 - 15 minutes until they are slightly golden.