Combine 1 1/2 cup white sugar, shortening, peanut butter, and brown sugar in a large bowl; beat with an electric mixer until creamy. Beat in eggs one at a time.
Mix flour, baking soda, baking powder, and salt together in a bowl. Stir gradually into the creamed shortening mixture until dough is combined. Chill dough until firm, about 2 hours.
Pour remaining white sugar into a shallow bowl.
Scoop up a portion of dough with a tablespoon. Press a chocolate-covered caramel into the center; cover up with dough and roll into a ball. Roll dough balls in sugar. Arrange on ungreased baking sheets.
Bake in the preheated oven until edges are firm and lightly golden, 9 to 12 minutes. Cool on the baking sheets for 5 minutes. Transfer to a wire rack to cool completely.