Preheat your oven to 350 degrees F. Grease a 9-by-5-inch loaf pan with nonstick cooking spray and set it aside. Whisk together the flour, cocoa powder, baking soda, and salt in a medium bowl. Set this dry mixture aside.
Add the eggs, melted butter, oil, sour cream, vanilla extract, and brown sugar in a large bowl. Stir until the mixture becomes smooth. Don't worry if you have a few small brown sugar clumps.
Stir the dry ingredients into the wet ingredients, careful not to overmix. Now, gently stir in the shredded zucchini until just combined. Add ¾ cup of the chocolate chips and mix them in.
Pour the batter into the prepared loaf pan. Sprinkle the remaining ¼ cup of chocolate chips over the top of the bread.
Bake for 50-60 minutes, or until a toothpick inserted into the center of the bread comes out mostly clean. Some melted chocolate on the toothpick is okay, but we don't want any gooey batter.
Remove the pan from the oven and set it on a wire cooling rack. Let the bread cool in the pan for 15 minutes. Afterward, run a knife around the edges of the bread and carefully remove it from the pan.
Allow the bread to cool on the wire cooling rack until slightly warm. Now you can cut it into slices and serve!