Preheat your oven to 350 degrees and grease a 9x13 glass pan, and set it aside.
Start with making your flax egg. Take a small cup and combine the 2 TB flax meal with 4 TB water stir until combined and set aside for 5 minutes to let it thicken.
Get the bowl of your stand mixer or just a large bowl. Add your dry ingredients; flour, baking powder, baking soda, salt, cinnamon, pumpkin pie spice, brown sugar, and white sugar, and stir until combined. Take a medium bowl and add all your wet ingredients; pumpkin, flax egg, oil, apple sauce, vinegar, and vanilla extract. Stir until combined.
Add your wet and dry ingredients with either your mixer paddle attachment or a hand electric mixer; mix until combined well but don't over-mix.
Spread evenly in the prepared 9x13 pan and cook for 30-37 minutes until a toothpick in the enter comes out clean. Place the pan on a cooling rack and cool completely before frosting.
Make your frosting; take a medium bowl and combine the frosting ingredients whip with the mixer until the frosting is light and fluffy. Frost the cake in an even layer.
Store the cake in the fridge in an airtight container; when you're ready to serve, let it come to room temperature and then serve.