Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan or line it with parchment paper.
In a large bowl, combine the mashed bananas, eggs, melted butter, granulated sugar, and vanilla extract. Mix well until smooth.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the banana mixture and stir just until combined. Do not overmix.
In a small bowl, mix the brown sugar and ground cinnamon to create the cinnamon sugar swirl.
Pour half of the banana bread batter into the prepared loaf pan and spread it evenly.
Sprinkle half of the cinnamon sugar mixture evenly over the batter layer.
Pour the remaining banana bread batter over the cinnamon sugar layer and spread evenly.
Sprinkle the remaining cinnamon sugar mixture on top of the batter.
Using a knife or skewer, gently swirl the cinnamon sugar into the batter to create a marbled effect.
Bake in the preheated oven for 55 to 65 minutes, or until a toothpick inserted into the center comes out clean.
Remove the bread from the oven and let it cool in the pan for 10 minutes.
Transfer the banana bread to a wire rack to cool completely before slicing and serving.