In a medium-sized mixing bowl, whisk together sugars, eggs, oil, applesauce, and vanilla extract.
In a large mixing bowl, whisk together flour, baking soda, baking powder, salt, and cinnamon.
Fold wet ingredients into dry ingredients until just combined.
Squeeze and blot grated zucchini with paper towels to remove extra moisture.
Fold zucchini into the batter.
Pour batter into a greased loaf pan.
Bake for 50-60 minutes or until a toothpick inserted into the center of the loaf comes out clean.
Be sure that the zucchini is finely grated. You don't want to find a big chunk of zucchini in the bread. Double check that the the food processor did its job before adding the zucchini to the batter.
Don't overmix. When you fold in the wet ingredients and when you add the zucchini, gently fold. If you overmix the batter, it can make the bread gummy.
I use a 9.25" x 5.25" loaf pan. Some loaf pans are not quite as wide and deep. If yours is a little smaller (9" vs 9.25"), you can still use it and you should be fine. If it's significantly smaller (8" or less), I suggest filling the pan 3/4 of the way to the top and making muffins with any extra batter. Keep in mind that smaller pans of batter take less time to bake.