Preheat oven to 350 ℉. Prepare 2-3 baking sheets with parchment paper or baking spray.
In a mixing bowl, mix together ½ cup salted butter, 1 cup white granulated sugar, and ½ cup brown sugar. Cream together until well combined, about 2 minutes. (You can use an electric mixer or do it by hand.)
Add in 1 large egg and 1 teaspoon vanilla, mix well for 2 minutes until smooth and creamy.
Add in 1 cup 100% pure pumpkin puree, 1 teaspoon cinnamon, ¼ teaspoon ginger, and ⅛ teaspoon allspice and mix well.
Add in 2½ cups all purpose flour,3 Tablespoons cornstarch, ½ teaspoon salt, 1 teaspoon baking powder, and ½ teaspoon baking soda . Mix until just combined and there are no flour pockets left. Dough will be very soft and sticky.
Scoop dough balls using a 1.5 Tablespoon scoop right into shallow dish of powdered sugar, and place on parchment paper or silicon mat lined baking sheet. Bake for 10-11 minutes, until cookies are puffy and cracked. Place a candy pumpkin in the center of each cookie while they're hot/warm. Can be right out of the oven.