Preheat oven to 350°F and spray a loaf pan with nonstick cooking spray.
In a medium bowl, whisk flour, baking soda, cinnamon, and salt. Set aside.
In a stand mixer with a whisk attachment or in a large bowl with a hand mixer, mix butter and sugar until light and fluffy, 2 minutes. Add eggs, one at a time, until incorporated. Add the bananas, applesauce and vanilla extract and mix to combine. Add dry ingredients and mix until no streaks remain. Fold in carrots and pecans until combined.
Pour into prepared pan and bake in preheated oven until a toothpick inserted in the center of the loaf comes out clean, 50-55 minutes. Remove from the oven and allow the bread to cool completely in the pan set on a wire rack.
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Notes
Storage: The bread will keep for around 3 days at room temperature or up to a week in the fridge in an airtight container.Freezing: This bread freezes really well, and will keep for up to 3 months. You can either freeze individual slices or the whole loaf. Wrap in parchment or plastic wrap and place in a freezer bag to freeze.