Preheat oven to 350 degrees F. Line two baking sheets with parchment paper.
To a large mixing bowl or mixer, beat the butter with a handheld or counter mixer until it reaches a light, fluffy consistency.
Add the brown sugar and granulated sugar, egg, vanilla extract, and pumpkin puree. Mix until thoroughly combined.
To a separate mixing bowl, combine the flour, baking powder, baking soda, salt, and pumpkin pie spice.
Slowly add the flour mixture to the wet ingredients 1/4-1/2 cup at a time. Mix until it reaches a cake batter consistency.
Using a 2-tablespoon scoop (ice cream scoop, melon scooper, or cookie dough scoop), release the batter onto the baking sheets, making sure to keep each cookie 2 inches apart.
Bake for 10-12 minutes, or until the tops are firm. Transfer the cookies to wire racks to cool before icing.
While the first batch is baking, combine the ingredients for the frosting into a medium mixing bowl or stand mixer and beat until smooth.
Spread frosting over each cookie and sprinkle with a cinnamon sugar mixture (optional). Store in an airtight container in the fridge.