In KitchenAid (or using an electric hand mixer) cream together butter, sugars, and pumpkin puree on medium-high speed, until well-combined
1 cup unsalted butter, 1 cup sugar, ½ cup brown sugar, ½ cup pumpkin puree
Add egg and egg yolk, beating until combined.Stir in vanilla extract.
1 egg + 1 egg yolk, 1 teaspoon vanilla extract
In separate, medium-sized bowl, whisk together flour, corn starch, pumpkin spice, and salt.
3 ⅓ cups all-purpose flour, 1 Tablespoon corn starch, 1 teaspoon baking powder, ¾ teaspoon pumpkin spice, ½ teaspoon salt
Gradually (with mixer on medium-low speed) stir the dry mixture into wet ingredients, pausing occasionally to scrape down the sides of the bowl to ensure that all ingredients are well-combined.
Transfer dough to refrigerator and chill 30-45 minutes.
Shortly before dough has finished chilling, preheat your oven to 350F and prepare cookie sheets by lining with parchment paper.
Remove dough from refrigerator and roll into heaping 1 ½ Tablespoon-sized balls.
Use the (clean) bottom of a glass or measuring cup to press down until cookies are about ½" thick.
Place on cookie sheet at least 2" apart and bake on 350F for 12-13 minutes.
Allow cookies to cool completely on cookie sheet before frosting.
Cream Cheese Frosting
In a stand mixer fitted with a paddle attachment (or using an electric mixer and a large bowl), beat together cream cheese, butter, and brown sugar until well-combined.
Stir in vanilla extract and salt, and then scrape down sides and bottom of bowl and beat briefly once more to ensure all ingredients are well-combined.
1 teaspoon vanilla extract, ¼ teaspoon salt
Gradually, with mixer on low-speed, stir powdered sugar into frosting until completely combined (pausing occasionally to scrape down sides and bottom of bowl).
3 cups powdered sugar
Once cookies have cooled completely, spread icing over the tops of cookies and immediately decorate with sprinkles.