Begin by placing the vegan butter and coconut oil in the bowl of a stand mixer (or large bowl). Use the paddle attachment or a hand mixer to beat the two together for 2 minutes until lighter in colour. Then add the brown sugar and beat for another 2 minutes to cream together. Be sure to scrape down the sides of the bowl as needed.
Once the sugar has dissolved, add the egg, pumpkin purée, and vanilla. Beat for another 2 minutes. The mixture will look slightly chunky.
Mix in the gluten-free all-purpose flour, almond flour, pumpkin pie spice, baking powder, and salt. Beat until just incorporated with no dry spots. The cookie dough will be the consistency of a thick cake batter and be quite sticky. Place the cookie dough in a freezer safe bowl, cover, and freeze for 1 hour. This will make the dough easier to handle.
Preheat the oven to 325°F and line 2 baking sheets with parchment paper. Remove the cookie dough from the freezer and use an ice cream scoop to scoop and shape 8 cookie dough balls. Place 4 on each baking sheet, spread apart. Press them down slightly into disks rather than balls. Bake for 20-22 minutes. Allow to cool for 5 minutes on the baking sheet before transferring to a cooling rack to cool completely.
While the cookies are cooling, make the icing. Place the vegan butter and cream cheese in the bowl of the stand mixer and beat with the paddle attachment (or use a hand mixer) for 2 minutes until combined and light in colour. Add the icing sugar and mix, starting on low. Once roughly incorporated, add the rest of the icing ingredients and beat for 2-3 minutes until smooth and airy.
Use a spoon or small scoop to place the icing on top of each cooled cookie. Use an offset spatula to spread the icing around the top of the cookie. Dust with cinnamon if you wish. Enjoy immediately or enjoy cold from the fridge.