Preheat the oven to 350°F (175°C). Spray an 8x4-inch loaf pan with nonstick spray (or grease it).
In a medium bowl, whisk together 1½ cups all-purpose flour, 1 teaspoon baking soda, ½ teaspoon baking powder, ½ teaspoon salt, and ½ teaspoon ground cinnamon. Set aside.
In a large bowl, whisk together ½ cup nonfat plain Greek yogurt, ⅓ cup honey, ⅓ cup packed light brown sugar, ¼ cup melted butter, ¼ cup unsweetened applesauce, 2 large eggs, and 1 teaspoon vanilla extract until smooth.
Mash 4 small (or 3 large) ripe bananas with a fork until mostly smooth, then whisk the mashed bananas into the wet mixture.
Add the dry ingredients to the wet ingredients and stir until just combined and no large streaks of flour remain—do not overmix.
Scrape the batter into the prepared loaf pan and smooth the top.
Bake on the center oven rack for about 55 minutes, or until the top is golden and a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the bread cool in the pan for at least 20 minutes, then turn it out onto a cooling rack and cool completely before slicing.