Sift the flour, baking soda, and salt into a medium bowl; set aside. Whisk the eggs and vanilla together in a liquid measuring cup; set aside. Lightly brush a 9 by 5 by 3 inch loaf pan with butter and preheat the oven to 350 degrees.
Next, cream the butter and sugar with a mixer until light and fluffy. Gradually pour the egg mixture into the butter while mixing until incorporated. Add the bananas.
With rubber spatula, mix in the flour mixture until just incorporated. Fold in the nuts and pour batter into pan. Bake for 55 minutes or until a toothpick inserted into the center of bread comes out clean.