Lightly grease a 5x5-inch baking dish with butter or cooking spray.
In a medium bowl, stir together the grated zucchini, granulated and brown sugars, applesauce (or pumpkin purée or melted butter), oil, egg, and vanilla until well combined.
Add the flour, baking powder, baking soda, cinnamon, and salt, then stir just until the flour disappears. The batter should be thick, so don't overmix.
Pour the batter into the prepared dish and smooth the top.
Bake for 30 minutes, until the top is set and a toothpick inserted in the center comes out clean.
Let the bread cool in the dish for 10 minutes, then transfer it to a wire rack to cool completely.
Notes
Use the right dish size. A 5-inch square dish (or a 4x6-inch dish) gives the loaf the depth it needs. A bigger pan spreads the batter thin so it bakes faster and can dry out, and a smaller, deeper one needs a few extra minutes.Mix the dry ingredients in gently. Whisk the wet ingredients until smooth, but once the flour goes in, stir only until it disappears. Overmixing works the gluten and leaves the loaf tough and dry.Start checking at 28 minutes. A small loaf can go from moist to dry quickly, so pull it as soon as a toothpick inserted in the center comes out clean.Use fresh leavening. Old baking powder or baking soda is the most common reason a quick bread won't rise and turns out dense. To test baking powder, add ¼ teaspoon to ½ cup of hot water; it should bubble. To test baking soda, stir ¼ teaspoon of vinegar into ½ cup of hot water, then add ¼ teaspoon of baking soda; it should fizz.If doubling the recipe, bake in either two 5x5-inch baking dishes or one 6x8-inch baking dish.