Prepare the flax egg by mixing 1 tablespoon ground flaxseed with 3 tablespoons water. Set it aside for several minutes until slightly thickened.
Add the room-temperature vegan butter, 1/2 cup granulated sugar, and brown sugar to a large mixing bowl. Beat with an electric mixer for about 2 minutes, until light and fluffy.
Add the prepared flax egg and alcohol-free vanilla. Mix for approximately 30 seconds until combined.
Add the all-purpose flour, pumpkin pie spice, baking powder, and salt. Mix until a thick cookie dough forms. The mixture may initially look crumbly, so continue mixing until it becomes soft and holds together.
Cover the bowl and refrigerate the dough for 30–60 minutes. The dough can also be refrigerated for up to 24 hours.
While the dough chills, prepare the pumpkin filling. Add the pumpkin puree, maple syrup, oat milk, cornstarch, vanilla, and pumpkin pie spice to a bowl.
Whisk until completely smooth and set aside.
About 10 minutes before removing the dough from the refrigerator, preheat the oven to 350°F (175°C). Line a large baking sheet with parchment paper.
Scoop the chilled dough into 10–11 evenly sized portions. Roll each portion between your palms to create a smooth ball.
Place the reserved 2 tablespoons of granulated sugar in a small bowl. Roll each dough ball in the sugar until lightly coated.
Arrange the dough balls on the prepared baking sheet and gently flatten each one with your palm.
Press your thumb into the center of each cookie. Carefully widen the indentation with your fingers, leaving a substantial border around the edge.
Spoon approximately 1 1/2 tablespoons of pumpkin filling into each indentation without covering the cookie edges.
Leave at least 2 inches between the Vegan Pumpkin Pie Cookies because they will spread while baking.
Bake for 12–14 minutes, until the cookie edges are lightly crisp and the pumpkin centers look set and slightly crinkled.
Cool on the baking sheet for 5 minutes. If necessary, gently reshape the edges with a spoon while the cookies are still warm.
Transfer to a wire rack and allow them to cool completely.
Once cooled, add a small swirl of vegan whipped cream and a light sprinkle of pumpkin pie spice, if desired.