Preheat oven to 350°F and line baking sheets with parchment paper for easy cleanup.
Cream together vegan butter, granulated sugar, and brown sugar until light and fluffy in bowl.
Beat in pumpkin and vanilla extract, mixing until smooth and everything is well combined together.
Whisk together flour, baking soda, baking powder, cinnamon, ginger, and salt in separate bowl completely.
Add dry ingredients to wet mixture, stirring gently until just combined without overmixing the dough.
Scoop dough onto prepared sheets, spacing cookies about two inches apart from each other evenly.
Bake cookies for 12-14 minutes until edges are set but centers still look slightly soft.
Cool on sheets for five minutes, then transfer cookies to wire rack to cool completely.
Beat frosting ingredients together until smooth, adding milk gradually to reach desired spreading consistency easily.
Frost cooled cookies generously with prepared frosting using offset spatula or butter knife evenly.