Preheat the oven to 350 degrees.
In a medium bowl combine the flour, baking soda, baking powder, pumpkin pie spice and salt. Set aside.
In a large bowl cream together the butter, brown sugar, and granulated sugar. Add in the pumpkin, egg, and vanilla and mix well.
Slowly pour the flour mixture into the butter mixture. Mix until well combined.
Using a #40 cookie scoop, scoop the batter onto the cookie sheets at least 2 inches apart.
Bake for 15-18 minutes or until the cookies look firm. Remove from oven and cool for 2 minutes before removing to a wire rack.
To make the frosting place the butter in a small saucepan. Allow it to cook over medium heat. Swirl the pan occasionally and check the color until it begins to brown. Once it starts to brown begin swirling it constantly until it's a golden brown and smells nutty. Remove from heat.
Add in the brown sugar, pumpkin pie spice, and vanilla. Mix well.
Slowly add the powdered sugar a half cup at a time, stirring well after each addition, until it is the desired consistency. Add 1-2 Tablespoons of milk if needed.
Spread the frosting on top of the cooled cookies.