Preheat oven to 375 degrees F and lightly butter 2 muffin tins.
In a large bowl, combine the flour, baking soda, and salt; set aside. Mash 2 of the bananas with a fork in a small bowl so they still have a bit of texture. With an electric mixer fitted with a wire whisk, whip the remaining bananas and sugar together like you mean it, for a good 3 minutes. Add the melted butter, eggs, and vanilla and beat well, scraping down the sides of the bowl once or twice. Mix in the dry ingredients just until incorporated. Fold in the nuts and the mashed bananas with a rubber spatula. Spoon the batter into the muffin tins to fill them about halfway. Give them a rap on the counter to get any air bubbles out.
Bake until a toothpick stuck in the muffins comes out clean, 18 to 20 minutes. Let cool for a few minutes before turning the muffins out. Serve warm or at room temperature.