Preheat oven to 350°. Prepare an 8x8 baking dish by lightly coating with non-stick oil spray.
In a large bowl, whisk together the flour, white sugar, baking soda, cinnamon, nutmeg, and salt.
In a separate bowl, mix together the melted vegan butter, sugars, vanilla, applesauce, non-dairy milk, and apple cider vinegar until combined.
Add the wet ingredients to the dry ingredients and mix until just combined.
Pour the batter into the prepared cake pan and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool for 15 minutes before inverting onto a wire rack to cool completely.
Put the powdered sugar in a sifter or a fine mesh sieve and dust the top of the cake, if you are using it. Serve and enjoy!
Notes
I use unsweetened soy milk but almond milk would work great too.
If you want a sweeter cake you can add a simple glaze. Simply mix a few tablespoons of powdered sugar with a small amount of non-dairy milk until you get a desired consistency. Then drizzle over the cooled cake.
Store leftover cake in an airtight container at room temperature for up to 3 days.