Preheat the oven to 350 degrees F. Spray a 9-inch square baking dish with cooking spray and set aside. Combine the cocoa, flours, baking powder, baking soda and salt in a bowl and set aside.
Melt the butter in a medium saucepan; set aside to cool slightly. Whisk in the sugar and 1/4 cup applesauce; using a spatula, mix the eggs in one at a time, followed by the vanilla; if the mixture appears too dry, add the remaining applesauce. Fold in the chocolate chips.
Pour the mixture into the prepared baking dish and bake until a cake tester inserted in the center comes out clean, 20 to 25 minutes. Cool for at least 20 minutes before cutting into squares.