Preheat your oven to 350°F (175°C). Line a standard 9×5-inch loaf pan with parchment paper or grease it well with butter and lightly dust with flour.
Mash the ripe bananas in a large mixing bowl until mostly smooth. A few small lumps are perfectly fine and will give the bread extra texture.
Add the eggs and sugar to the mashed bananas and whisk until combined. Pour in the melted butter and stir until the mixture is smooth.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Gradually fold the dry ingredients into the banana mixture using a spatula. Do not overmix. Stir just until no dry flour remains. Overmixing develops gluten and can make the loaf dense instead of soft and tender.
Pour the batter into the prepared loaf pan and smooth the top. Bake for 45–60 minutes, or until a toothpick inserted into the center comes out clean or with just a few moist crumbs. The top should be golden brown with a beautiful crack running through the center—that's exactly what you want.
Let the banana bread cool in the pan for about 10–15 minutes, then transfer it to a wire rack to cool completely. Resist slicing it while it's still hot—the texture continues to set as it cools.