Preheat the oven to 400 degrees. Prepare two large rimmed baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and pumpkin spice.
Beat the butter and the sugars on medium speed, scraping down the sides as needed, beating for 5 to 7 minutes or until fluffy.
Add the egg and pumpkin puree, beating until evenly combined. Add the flour mixture and beat until mixed.
Using a small (1 tablespoon) cookie scoop, portion the cookie dough and shape into balls. Place on the baking sheet about 1 inch apart.
Bake for 8 to 10 minutes or until the cookies are puffed and the edges are set.
Remove the cookie sheets from the oven and immediately press one candy into the center of each cookie.
Transfer to a wire rack and allow the cookies to cool completely. The candies will look like they are melting but do not disturb them and they will resolidify in the correct shape.
Store in an airtight container for up to 1 week or refrigerate for up to 3 weeks. Cooled cookies can also be frozen but sudden temperature changes may cause the candy to “bloom” which is harmless but not pretty.