Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a large bowl, mash the ripe bananas until smooth.
Add the eggs, unsweetened applesauce, Greek yogurt, and vanilla extract to the mashed bananas. Mix well until combined.
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.
If using, fold in the chopped walnuts.
Pour the batter into the prepared cake pan and spread evenly.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.