Preheat oven to 350°F. Grease a 9x5 inch loaf pan or line with parchment paper.
In a large bowl, mash the bananas with a fork until mostly smooth. Stir in the brown sugar and applesauce until blended.
Mix in the eggs and vanilla extract.
Add the flour, baking soda, cinnamon, and salt on top of the wet mixture. Fold together with a spatula until the flour just disappears. Do not overmix.
Pour the batter into the prepared loaf pan and spread evenly.
Bake for 55 to 65 minutes, until a toothpick inserted in the center comes out clean. If the top browns too quickly, tent with foil for the last 15 minutes.
Cool in the pan for 10 minutes, then transfer to a wire rack to cool completely before slicing.
Notes
Banana ripeness: The browner the banana, the better. Fully brown or black bananas are sweeter and mash easily.
Don't overmix: Fold just until the flour disappears. Overmixing makes the bread tough.
Sweetened applesauce: If using sweetened applesauce, reduce brown sugar to ½ cup.