Preheat the oven to 350ºF (180ºC). Grease a 9x13-inch (23 x 33 cm) baking pan and line it with parchment paper.
In a small bowl, mix the ground flaxseed and water. Let it sit for 5-10 minutes, or until thickened.
In a large bowl, whisk together the flour, baking powder, baking soda, salt, pumpkin pie spice, and cinnamon.
In another large bowl, whisk together the pumpkin puree, brown sugar, granulated sugar, oil, vanilla extract, and flax mixture until well combined.
Pour the wet ingredients into the bowl with the dry ingredients. Stir until just combined, being careful not to overmix.
Pour the batter into the prepared pan and spread it out evenly with a spatula.
Bake for 30-35 minutes, or until the center is set and a toothpick inserted into the middle comes out clean or with a few moist crumbs.
Let the bars cool completely before spreading the vegan cream cheese frosting on top, if using. You can serve them right away, or refrigerate them for about 30 minutes first to help the frosting set and make the bars easier to slice cleanly.
Notes
Store frosted bars in an airtight container in the refrigerator for up to 5 days.
Freeze the bars in a freezer-safe container for up to 3 months. For the best texture, freeze them without the frosting and add it after thawing.
Thaw frozen bars overnight in the refrigerator before serving.