Preheat the oven to 350°F (175°C). Grease a standard 9×5-inch loaf pan or line it with parchment.

Peel and mash the bananas in a large bowl until mostly smooth with a few small lumps.

Stir the sugar into the mashed bananas until combined. Pour in the melted butter and mix until even.

Beat the eggs in a small bowl. Add them to the banana mixture and stir until incorporated. Mix in vanilla if using.

In another bowl, whisk together the flour, baking soda, and salt so they are evenly distributed.

Add the dry mix to the wet mix in batches. Fold gently with a spatula until just combined. A few small lumps are fine—do not over-mix.

Pour the batter into the prepared loaf pan and smooth the top with a spatula.

Bake in the center of the oven for 50 to 60 minutes. Start checking at 50 minutes; a toothpick or skewer inserted in the center should come out clean or with a few moist crumbs.

Let the bread cool in the pan for 10 to 15 minutes, then transfer to a rack to cool completely before slicing.