In KitchenAid (or using an electric hand mixer) cream together butter, sugars, and pumpkin puree on medium-high speed, until well-combined
Add egg and egg yolk, beating until combined.Stir in vanilla extract.
In separate, medium-sized bowl, whisk together flour, corn starch, pumpkin spice, and 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.
Gradually, with mixer on low-speed, stir powdered sugar into frosting until completely combined (pausing occasionally to scrape down sides and bottom of bowl).
Once cookies have cooled completely, spread icing over the tops of cookies and immediately decorate with sprinkles.