Preheat the oven to 350°F (175°C). Lightly grease a 9×5-inch loaf pan or line it with parchment paper.
In a large mixing bowl, combine the mashed bananas, vegetable oil, granulated sugar, brown sugar, and vanilla extract. Mix well until smooth.
In a separate bowl, whisk together the all-purpose flour, baking soda, salt, and ground cinnamon.
Add the dry ingredients to the wet ingredients and stir gently until just combined.
Add the unsweetened applesauce and non-dairy milk to the batter and mix until incorporated. Do not overmix.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake in the preheated oven for 55 to 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove the banana bread from the oven and allow it to cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing and serving.