Preheat the oven to 350 degrees F and line a cookie sheet with parchment.

Whisk together the flour, baking powder, baking soda, pumpkin pie spice, and salt in a medium-size bowl.

In a stand mixer or with an electric mixer, cream together the butter and sugar until light and fluffy, about 2 minutes. Then add the egg, pumpkin puree, and vanilla; stir until combined.

Gradually fold the dry ingredients into the pumpkin butter mixture. Using a cookie scoop, transfer mounds of pumpkin cookie batter to the prepared cookie sheet. Two spoons work well too. Bake in the preheated oven for 17 to 20 minutes or until lightly browned.

Cool on a wire rack and make the icing.

To Make the Icing: In a small saucepan, bring 3 tablespoons of butter and 4 tablespoons of milk to a boil. Stir in 1/2 cup of brown sugar and 1/2 teaspoon of vanilla. Let the mixture cool and then stir in powdered sugar until you reach spreading consistency.