Heat the oven to 350°F. Grease an 8-by-8-inch square pan or loaf pan with baking spray. Line with parchment paper long enough to extend up the sides to use as "handles" for removing the bread once baked.
Grate the zucchini on the medium side of the box grater.
In a large bowl, whisk the eggs, brown sugar, oil, molasses, and vanilla until smooth.
In a separate large bowl, whisk the flour, salt, baking soda, and baking powder.
Add the dry ingredients to the wet ingredients and fold together with a silicone spatula until just combined.
Add shredded zucchini and chocolate chunks; stir until just combined.
Pour the batter into the prepared pan. If desired, top with a few extra chocolate chunks.
Bake the bread for 40 to 45 minutes, until a toothpick inserted in the center comes out clean.
If using an 8-by-8-inch pan, cool the bread completely in the pan on a wire rack. If using a loaf pan, cool the bread for 20 minutes, then use parchment paper handles to remove it from the pan and transfer it to a cooling rack to cool completely.