Cream shortening, peanut butter, and both sugars in a large bowl with an electric mixer until light and fluffy. Beat in egg until incorporated. Mix flour, baking soda, baking powder, and salt in a separate bowl; gradually stir into peanut butter mixture until combined. Cover and refrigerate dough for at least 1 hour.
Preheat the oven to 375 degrees F (190 degrees C).
Roll chilled dough into walnut-sized balls and place 2 inches apart onto baking sheets. Flatten dough balls with a fork dipped in flour to form a crosswise pattern.
Bake in the preheated oven until edges are set, 10 to 12 minutes. Transfer cookies to a wire rack to cool completely.