Prepare the flax egg by mixing 1 tablespoon ground flaxseed with 3 tablespoons water; set aside.
In a mixing bowl, cream together vegan butter, granulated sugar, and brown sugar until smooth. Add the flax egg and vanilla; mix well.
In another bowl, whisk together flour, pumpkin pie spice, baking powder, and salt. Gradually add to the wet mixture until combined.
For the filling, mix pumpkin puree, maple syrup, oat milk, cornstarch, vanilla, and additional pumpkin pie spice until smooth.
Preheat oven to 350°F (175°C). Scoop cookie dough onto a parchment-lined baking sheet; create an indentation in each ball and fill with pumpkin mixture.
Bake for about 15 minutes until edges are golden. Cool slightly before serving.