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