Preheat the oven to 350°F (175°C) and line a 12-cup muffin tin with paper liners.
In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.
In a separate bowl, combine the mashed bananas, granulated sugar, beaten egg, melted unsalted butter, and vanilla extract. Mix well.
Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix; the batter should be slightly lumpy.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 18 to 20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and let them cool in the pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.