Preheat the oven to 350℉. Grease a 9" x 5" loaf pan, then line with parchment paper to form a sling.
In a large bowl, whisk the flour, baking powder, baking soda, salt, cinnamon, and allspice. Make a well in the center of the bowl, and add the oil, agave, maple syrup, granulated sugar, brown sugar, applesauce, orange juice, water, and vanilla extract. Whisk well, ensuring that the ingredients are well combined and the batter is smooth.
Pour the batter into the prepared pan. Bake for 60-65 minutes, or until a tester comes out clean or with just a few dry crumbs.
Allow the cake to cool in the pan for about 20 minutes, then, using the parchment paper as a sling, carefully lift it out of the pan onto a wire rack.
When the cake is fully cool, prepare the icing (if using). In a small bowl, whisk together the powdered sugar, non-dairy milk, and orange juice until smooth. Use a spoon to drizzle the icing over the loaf cake. Allow the icing to harden, then slice and serve.