Preheat the oven to 350°F and line a couple of baking sheets with parchment paper.
Whisk the flour, baking powder, baking soda, pumpkin pie spice, cinnamon, and salt together in a large mixing bowl. Set this aside.
Place the vegan butter, brown sugar, sugar, molasses, and vanilla into another large mixing bowl. Use an electric mixer to cream the ingredients together until fully blended.
Beat the pumpkin puree and milk into the butter mixture.
Using the mixer at low speed, begin beating the flour mixture into the pumpkin mixture, adding about a third at a time and beating each addition in until fully incorporated before adding more.
Drop the dough by rounded heaping tablespoonfuls onto the prepared baking sheets, leaving at least an inch between cookies.
Bake the cookies for about 12 minutes, or until they're puffed up quite a bit and spread out slightly.
Place the baking sheets on cooling racks. Let the cookies cool for a few minutes, until they feel fully set, then remove them from the baking sheets and place them directly on the cooling racks to finish cooling.
Once the cookies are cool, stir the icing ingredients together in a medium bowl. You can add more milk if the mixture seems too thick, or more powdered sugar if it seems too thin.
Dip the tops of the cookies in the icing to coat them, then place the cookies back in the racks until the icing has set. If you'd like, you can top each cookie with a sprinkle of cinnamon as well.
Serve.