First, preheat the oven to 400F. Spray a 9x5 loaf cake with nonstick spray, or coat with vegan butter and flour. Set aside.
Next, prepare your flax egg. Mix the flaxseed with the water, and set aside while you prepare the rest.
Now, grate your zucchini, and place in a kitchen towel. Wrap the towel up tightly, and squeeze over the sink to get rid of all the excess water. This is essential so that your zucchini bread won't be a soggy mess!
Place your squeezed zucchini in a large bowl, and mix in the milk, oil, flax mixture, vanilla, cinnamon, and brown sugar.
Mix everything together, then add in the dry ingredients - flour, baking powder, baking soda, and salt. Batter will be thick!
Pour half of the batter into the pan, then sprinkle about half of the swirl mixture on top (½ tablespoon brown sugar and ½ teaspoon cinnamon).
Pour the other half of the batter on top, then the rest of the cinnamon swirl mix. Using a knife, make swirls throughout the bread to help the cinnamon and brown sugar get mixed in. Don't swirl too much! This will ensure you have those little pockets of sweetness in the bread, and spots of caramelized sugar on top (the best part!).
Bake in preheated oven at 400F for about 5 minutes, then lower heat to 350F, and bake for 35-40 more min. Don't open the oven or take the bread out when you turn down the temperature.
Let bread cool slightly, then remove from pan to finish cooling completely. Be careful letting it cool all the way in the pan, as the cinnamon sugar swirl will solidify and can make your bread stick to the pan!