Grease and flour a 9x5x3 baking pan. Preheat oven to 350 degrees F and place rack at center.
Mix the dry ingredients (flours, cornmeal, baking soda, and salt) in a medium mixing bowl and set aside.
Add the bananas, eggs, sugar, butter, and applesauce to the bowl of food processor fitted with a steel blade. Pulse several times until the mixture is well-combined, but thick and chunky; it needn’t be smooth.
Scrape the banana mixture into the bowl with the dry ingredients along with the squeezed-dry zucchini and briefly mix. In a cup, stir together the chocolate chips with the reserved teaspoon of flour. Add the floured chocolate chips to the bowl and mix until just barely combined. Do not overmix.
Spoon the bread mixture evenly into prepared pan, tapping the pan on the counter to settle the batter and then smoothing the top. Place pan on a rimmed baking sheet to avoid overbaking the bottom of the bread. Bake 60-70 minutes or until a skewer inserted at the center of the bread comes out clean. Remove from oven and let cool in pan for 5-10 minutes. Holding the sides of the pan with potholders or a towel, gently but firmly bang the bottom of the pan onto a board or counter. If the bread hasn't loosened, run a thin knife around the sides of the pan. Turn out onto a rack and cool completely–at least an hour or two.STORAGE: Wrap well with plastic wrap and store at room temperature for 1-2 days or refrigerate for a week. If you're in a really warm, humid climate, you might want to refrigerate your bread soon after it's cool. FREEZING: Wrap well first in plastic wrap and then in foil and freeze for up to 3 months.