Heat oven to 375˚F/190°C. Remove wrappers from caramel kisses.
Cream together the butter and shortening. Add 1/2 cup of brown sugar and sugar. Beat well. Add the egg, salt, pumpkin pie spice, baking powder, and pumpkin puree. Mix until combined. Gradually add the flour while mixing until all the flour is incorporated.
Scoop out dough by the tablespoon. Mold dough around each caramel kiss, covering completely. Mix remaining 1 cup of brown sugar and cinnamon together. Roll dough balls in brown sugar and cinnamon mixture. Place cookies 2-inches apart on cookie sheet lined with parchment paper or a silicone mat.
Bake for 10-12 minutes or until cookies are set. Remove from oven and place cookie sheet on wire rack. Allow cookies to cool completely.
Prepare Chocolate Drizzle by placing white chocolate morsels in a small microwave safe bowl. Microwave at medium (50%) for 30 seconds; stir. Continue to microwave at medium (50%) at 10 second intervals, stirring well after each heating, until chocolate is melted and mixture is smooth when stirred.
To drizzle over cookies, pour melted chocolate into a ziptop bag. Snip the tip off a bottom corner of the bag and gently squeeze the chocolate over the cookies. Let set.