Preheat oven to 350F and lightly oil and flour an 8-inch square baking dish. In a small bowl, whisk together the flax seeds and warm water. Set aside.
In a large mixing bowl, whisk together the flours, sugar, cocoa powder, baking powder, and salt.
In a medium sized mixing bowl, whisk together the non-dairy milk, espresso/coffee, oil, applesauce, vanilla, and the flaxseed mixture. Add the wet ingredients to the dry ingredients and mix until they're combined. Fold in the chocolate chunks or chips, reserving a few to sprinkle on top of the brownies.
Bake for 25-35 minutes, or until a toothpick inserted into the brownie center emerges with just a couple crumbs. Start checking the brownies at the 22 minute mark, just to be safe: if they're burning at the edges or getting very firm in the middle, use a toothpick to check. You want to avoid overcooking for a perfectly moist, chewy texture!
Transfer the baking dish to a wire cooling rack and allow the brownies to cool for 30 minutes before cutting and serving.