Beat shortening, 1 cup white sugar, and light brown sugar together in a bowl until light and fluffy. Stir in pumpkin puree; beat in eggs and vanilla extract.
Whisk flour, baking powder, 1 1/2 teaspoons cinnamon, cream of tartar, salt, and nutmeg together in a bowl. Gradually stir flour mixture into pumpkin mixture until dough is just-combined. Cover the bowl with plastic wrap and refrigerate until chilled, at least 1 hour.
Preheat oven to 350 degrees F (175 degrees C). Line baking sheets with parchment paper.
Whisk 1/2 cup white sugar, 1 teaspoon cinnamon, and allspice together in a small bowl. Roll dough into 1-inch balls; roll balls in cinnamon-sugar mixture and place 2 inches apart on prepared baking sheets. Slightly flatten each ball with a flat-bottomed glass.
Bake in the preheated oven until golden and set, 12 to 13 minutes. Cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.