Preheat the oven to 350°F. Line two baking sheets with parchment, silicone mats or use non-stick sheets.
Whisk the dry ingredients (GF flour, baking powder, baking soda, salt, cinnamon, & pumpkin pie spice) together in a small bowl. Set aside.
Place the butter and brown sugar in the bowl of a stand mixer. Cream together until light and fluffy, 1-2 min.
Add in the pumpkin, egg, vanilla, and milk. Mix until well combined.
Add in the dry ingredients and mix again until well combined.
Drop by tablespoonful onto prepared cookie sheets. Bake 10-12 minutes or until bottoms are lightly golden and cookies spring back when tops are touched. Allow to cool completely before frosting.
For the Frosting
Cream the softened butter and cream cheese together.
Add in the vanilla and powdered sugar. Mix until well combined.
Add in 1 Tablespoon of milk and mix. Continue to add milk by the tablespoon and mix until the frosting has a smooth, spreadable consistency.
Frost the cooled cookies and sprinkle with cinnamon or pumpkin pie spice if desired.