Preheat oven to 350°F (175 °C) and line a 9 x 5-inch loaf pan with parchment paper.
To a large mixing bowl add the applesauce, oil, dark brown sugar, dairy-free milk, cinnamon, ginger, nutmeg, salt, and vanilla. Mix to combine.
Add the flour, baking powder, and baking soda. Fold together until just combined. Don't overmix. Make sure to scrape right into the bottom of the bowl as the flour likes to hide out down there.
Spoon into the lined pan. I like to sprinkle granulated sugar over the top for extra texture, but if you're icing the cake, this isn't essential. Bake for 55 to 60 minutes, or until the cake is well-risen, golden, starting to come away from the sides of the pan, and a skewer or toothpick inserted into the middle comes out clean. Dark/black pans cook quicker, so if using one, start checking a little earlier.
Remove from the oven. Leave in the pan for 5 minutes, then lift out and onto a wire rack. Carefully remove the parchment paper and allow to cool completely before making the icing.
For the icing (this cannot be made in advance)
Melt the butter, brown sugar, cinnamon, and milk together in a small saucepan over medium-low heat. Whisk or stir until the sugar has dissolved, then turn off the heat.
Add the powdered sugar to the pan and whisk or stir until completely smooth.
Drizzle the icing generously over the cake, letting it run down the sides. It will take about 10 minutes for the icing to set.
NOTES
STORING - Because of the high moisture content, this cake doesn't do well in a sealed container. While uncut and un-iced, you can leave it sitting on a plate or cutting board draped with a clean tea towel. Once cut, I find it best to store it cut side down on a wooden cutting board or plate in my pantry (or a cupboard).