Preheat the oven to 350°F. Grease a 9 x 5-inch loaf pan or line it with parchment paper.
Place the bananas in a large bowl and mash until mostly smooth.
Add the sugar and melted butter. Whisk until combined.
Add the eggs one at a time, mixing after each addition. Stir in the vanilla extract.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt.
Add the dry ingredients to the banana mixture. Fold gently just until no dry streaks remain. Do not overmix.
Pour the batter into the prepared pan and smooth the top. If desired, make a shallow line lengthwise through the center of the batter with a buttered knife.
Bake for 50 to 55 minutes, or until a toothpick inserted near the center comes out clean or with a few moist crumbs.
Allow the bread to cool in the pan for 15 minutes. Transfer it to a wire rack and let it cool for at least 30 additional minutes before slicing.