Preheat the oven to 425 degrees F. Lightly brush a 12-muffin tin with butter and set aside.
Whisk the flour, baking soda, brown sugar, and cinnamon together in a medium bowl; set aside.
Whisk the banana, oil, milk, eggs, salt, and vanilla in a large measuring cup with a spout or another bowl.
Make a small well in the center of the dry ingredients. Pour wet ingredients into the center; then stir with a wooden spoon until the dry ingredients are moistened but still lumpy. Do not overmix the batter or your muffins will come out dense. Gently stir in the nuts. Divide the batter evenly into the muffin tin.
Put the muffins in the oven and immediately reduce oven temperature to 375 degrees F. Bake until golden brown, about 25 minutes, rotating the pan halfway through the cooking. (Insert a toothpick into the center of a muffin to check if it is done. Toothpick should come out clean). Cool muffins in the pan on a rack for a couple minutes. Turn the muffins out of the pan and cool on the rack. Serve warm or at room temperature.
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