In a mixing bowl, beat butter and sugars until light and fluffy (about 2 minutes).
Add dulce de leche to a piping bag with a narrow decorating tip. This will make it much easier to drizzle it on cookies. Set aside.
Add egg to the butter and sugars, beat well, and scrape sides and bottom of the bowl.
Add pumpkin puree and Nutella, and mix well. Scrape sides and bottom of the bowl.
Bring the speed to low, and sift in flour, salt, and baking soda, and beat until combined. Scrape sides and bottom of the bowl, and mix a few more times.
Refrigerate dough for an hour.
Preheat the oven to 350 degrees. Line a large baking sheet with parchment paper.
Scoop cookie dough and roll into about 1-inch balls with your hands. Flatten slightly and place them on the prepared baking sheet, and repeat with remaining dough. Set cookies about 2 inches apart, and bake for 9–11 minutes. (If you have to bake cookies in batches, refrigerate dough in between.)
Cool on a wire rack for about ten minutes and drizzle with dulce de leche.