Preheat the oven to 350F and grease a standard 9x5 inch loaf pan, lining the bottom with parchment paper.
Peel the large ripe bananas and place them in a large mixing bowl. Mash them thoroughly with a fork until they are mostly smooth.
Pour the melted unsalted butter into the mashed bananas and stir well to combine.
Mix the brown sugar, plain yogurt and vanilla extract into the banana mixture until everything is evenly incorporated.
Using plain yogurt instead of eggs is the secret trick here, keeping the bread incredibly soft and moist without changing the flavor. Sprinkle the all-purpose flour, baking soda and salt over the wet ingredients.
Fold the dry ingredients in gently using a spatula, stopping as soon as you no longer see any streaks of dry flour.
Pour the thick batter into your prepared loaf pan and smooth the top evenly.
Bake for 50 to 60 minutes. Insert a toothpick into the center, and if it comes out clean, the bread is done. Let it cool before slicing.
