Grease a standard size loaf pan (9"x5"). Set aside.
In a medium bowl whisk together the flour, baking soda, salt, and cinnamon. Set aside.
With a stand or handheld mixer, beat the softened butter and brown sugar on high speed until smooth, about 2-3 minutes. Beat eggs in, one at a time; beating well after each addition. Add applesauce and beat until well incorporated. Add the mashed banana and vanilla, beating just until combined.
With the electric mixer on low, slowly add dry ingredients. Continue mixing just until combined being careful to not over-mix. Fold in nuts, optional.
Pour the batter into a prepared bread pan. Bake in center of oven for 55-60 minutes. If the top is browning too fast, lightly lay a piece of foil over the top of the pan. When a toothpick inserted into the center of the loaf comes out clean, with only a few crumbs, it's ready to take out of the oven. Make sure to keep an eye on the banana bread during the last 10 minutes or so of baking time. Every oven is different and you definitely don't want this bread to overbake!
Remove from oven and allow to cool 60 minutes, in the pan, on a wire rack. When the hour is up, remove the banana bread from the pan and place back on wire rack to finish cooling.