Preheat oven to 350 degrees. Use parchment paper or a silpat mat to line a baking sheet.
Cream the butter, brown sugar and granulated sugar in a stand mixer until light and fluffy (about 2-3 minutes). Scrape the sides of the bowl to make sure all ingredients are blended.
Add in egg yolk, vanilla and pumpkin puree and beat until combined.
Add in flour, baking powder, salt, pumpkin pie spice, and cinnamon. Mix until all combined.
Scoop out dough and roll into balls. Scoop out 1-2 TBSP dough for smaller cookies (or 1/4 cup dough for BIG cookies). Place dough balls on prepared pan (12 cookies to a pan if smaller cookies) and slightly flatten with your hand to about 1/2 - 1 inch thick.
Bake for 8-9 minutes. Cool cookies cookie sheet for 10 minutes before removing to cooling rack. NOTE: If you decide to make the bigger cookies, you'll need to bake them for 12-13 minutes.
While cookies are cooling, make the frosting.
In stand mixer, cream butter and cream cheese for about 2-3 minutes until smooth.
Add in the vanilla, powdered sugar, and caramel and mix on high for about 3 minutes until smooth.
Put the frosting in a piping bag with a star tip OR you can put the frosting in a gallon size Ziploc bag and cut off one corner of the bag. Use the frosting bag to pipe frosting in a spiral on top of the cooled cookies.