In a medium bowl, whisk together the flour, baking powder, baking soda, pumpkin pie spice, cinnamon and salt. Set aside.
In a large bowl, beat the dairy free butter, brown sugar and granulated sugar for 2 minutes until light and fluffy. Beat in the pumpkin puree, egg and vanilla until combined, even if the mixture looks slightly separated.
Add the dry ingredients and mix on low speed just until no streaks of flour remain. Cover the bowl and chill the soft dough for 30 minutes so the cookies bake up thick and pillowy.
Preheat the oven to 350°F and line two baking sheets with parchment paper. Scoop 2 tablespoon mounds of dough about 2 inches apart and gently flatten the tops with damp fingers, since these cookies spread very little.
Bake for 12-14 minutes, until the tops spring back when lightly touched. Cool on the sheet for 5 minutes, then transfer to a wire rack to cool completely.
For the frosting, beat the dairy free cream cheese and dairy free butter until smooth. Add the powdered sugar, vanilla and cinnamon and beat just until creamy and fluffy, about 1 minute.Avoid overbeating, since dairy free cream cheese can turn runny. If the frosting feels soft, chill it for 10 minutes before using.
Spread a generous swirl of frosting over each cooled cookie. Store in an airtight container in the fridge for up to 4 days.