Preheat the oven to 350°F and lightly grease an 8×8-inch baking dish.
In a small bowl, mix 1/4 cup ground flaxseed with 1/4 cup plus 2 tablespoons warm water. Let it sit for 5 minutes to thicken.
In a large bowl, combine 3/4 cup all-purpose flour, 3/4 cup almond flour, pumpkin pie spice, baking powder, baking soda, and salt.
In a medium bowl, whisk together 1 cup canned pumpkin, 2 tablespoons melted coconut oil, 1/3 cup maple syrup, 1 teaspoon vanilla extract, and the flaxseed mixture.
Pour the wet ingredients into the bowl of dry ingredients and stir until just combined. Spread the batter into the baking dish (it will be thick) and bake for 25 to 30 minutes or until a toothpick comes out clean. Let the cake cool completely before frosting.
While the cake cools, make the frosting: In a large bowl, beat 4 ounces vegan cream cheese and 1 1/2 tablespoons vegan butter with an electric mixer until smooth**. Add 1/8 teaspoon vanilla and 2 cups powdered sugar, mixing until smooth. Spread over the cooled cake, sprinkle with chocolate chips (if desired), and slice into bars.
