To make the frosting, soak the cashews in warm water for 3-4 hours. If you don't have time, soak the cashews in scolding hot water for 15 minutes.
Place soaked cashews in a food processor or blender until a paste forms and no visible chunks of cashews are left. Scrape down the sides then add the coconut oil, maple syrup, juice, vanilla and pumpkin pie spice. Blend on high until combined then transfer to the refrigerator until ready.
To make the cookies, in a medium bowl, whisk together the almond butter, pumpkin, egg, egg white, coconut sugar and vanilla until combined. In a separate small bowl, whisk together the almond flour, pumpkin pie spice, baking soda and salt. Add the dry mixture to the wet mixture and whisk until no clumps are left. Place in the refrigerator to chill for 30 minutes.
Preheat oven to 350°F and line a large baking sheet with parchment paper or a silpat mat. Once dough is chilled, remove from the refrigerator and scoop heaping tablespoons onto the prepared baking pan. The dough will be thick and somewhat wet and sticky. Gently roll/press the dough until it's 1 inch thick and flatter on top. Cookies won't spread in the oven.
Bake for 8-10 minutes, or until cookies are set. Remove from oven and cool on baking sheet for 5 minutes then move to wire baking rack to cool completely. Spread 1-2 tablespoons of frosting on each cookie once they're completely cooled. Store in the refrigerator for up to one week.