Sift flour, baking soda, and salt together in a small bowl.
Beat shortening, white sugar, brown sugar, rum, and vanilla extract together in a separate bowl. Add eggs one at a time, beating well after each addition. Stir in white chocolate and butterscotch chips. Fold in the flour mixture until just combined. Refrigerate dough until firmed up slightly, about 30 minutes.
Preheat the oven to 375 degrees F (190 degrees C). Place a baking stone in the oven.
Drop dough by rounded tablespoonfuls onto the preheated baking stone. Bake until cookies start to turn golden, about 11 minutes. Remove from oven and let cookies rest on the stone for a few minutes. Transfer them onto a wire rack to cool completely.