Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a large bowl, combine the mashed bananas, eggs, unsweetened applesauce, vegetable oil, and vanilla extract. Mix until smooth.
In a separate bowl, whisk together the whole wheat flour, baking soda, baking powder, ground cinnamon, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared cake pan and spread evenly.
Bake for 35 to 40 minutes or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack and allow it to cool completely before serving.