Preheat oven to 350 degrees F (175 degrees C). Spray 2 muffin tins with cooking spray or line with paper liners.
Whisk applesauce, mashed bananas, butter, and eggs together in a large bowl.
Combine flour, white sugar, brown sugar, cinnamon, vanilla extract, baking soda, and salt in a separate bowl. Fold into the applesauce mixture until batter is just combined.
Divide batter evenly among muffin tins, filling each cup almost full.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 15 minutes.
