Preheat oven to 350°F. Spray an 8x8 baking dish with non-stick cooking spray.
In a medium bowl, mash 2 very ripe bananas until mostly smooth.
Add 1/2 cup granulated sugar, 1/4 cup brown sugar, 1/2 cup applesauce (no sugar added), 1/3 cup water, and 1 teaspoon white vinegar to the mashed bananas. Stir until thoroughly combined.
Add 1 1/4 cups all-purpose flour, 1/4 cup unsweetened baking cocoa, 1 teaspoon baking soda, and 1/4 teaspoon salt to the wet mixture. Stir just until the dry ingredients are incorporated and no large streaks of flour remain (do not overmix).
Fold in 1/3 cup of the 2/3 cup semi-sweet chocolate chips.
Pour the batter into the prepared 8x8 baking dish and spread evenly.
Sprinkle the remaining 1/3 cup semi-sweet chocolate chips evenly over the top of the batter.
Bake for 33–37 minutes, or until a toothpick inserted into the center comes out clean or with just a few moist crumbs.
Remove from the oven and let the cake cool in the pan before slicing and serving.