Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a large mixing bowl, combine the mashed bananas, eggs, unsweetened applesauce, whole milk, and vanilla extract. Mix well until smooth.
In a separate bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cinnamon, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the melted unsalted butter until evenly incorporated.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 30 to 35 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 serving to toddlers.