Preheat the oven to 350°F (175°C) and grease your 8x8 inch baking pan.
In a large mixing bowl, whisk together the melted butter and sugar until well combined and slightly glossy.
Add the eggs one at a time, whisking after each addition, until fully incorporated. Mix in the mashed bananas and vanilla extract until the mixture is smooth.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add these dry ingredients to the banana mixture, folding gently with a spatula until just combined and no streaks remain.
Pour the batter into the prepared baking pan and level the surface with a spatula. Bake in the preheated oven for about 30 minutes, or until the top is golden brown and a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes. Then transfer to a wire rack to cool completely before slicing and serving.