Preheat the oven to 350°F (175°C) and line an 8x8 inch baking pan with parchment paper.
Shred the zucchini on the fine side of a box grater. Do not squeeze out the liquid. That moisture is what replaces the oil and keeps these brownies fudgy.
In a large bowl, whisk the egg, egg white, applesauce, monk fruit sweetener and vanilla until smooth. Stir in the shredded zucchini.
Add the flour, cocoa powder, baking soda and salt. Fold until no dry streaks remain. The batter will look thick and a little dry at first.
Let it rest for 5 minutes, then stir again. The zucchini will release its moisture and the batter will loosen into a thick, spreadable consistency. Fold in the chocolate chips. Spread the batter into the pan, pressing it into the corners with a spatula.
Bake for 25-30 minutes, until the top is set and a toothpick inserted in the center comes out with moist crumbs.
Cool completely in the pan before cutting into 16 squares. These are even fudgier on day two.