Heat oven to 350°F. Spray muffin tins (12 cup) with nonstick cooking spray.
In a large bowl, mash bananas (see note). Whisk in eggs until combined. Add applesauce, oil, sugar, and vanilla extract. Stir well until combined.
Add flour, baking soda, baking powder, and salt on top of banana mixture. Mix the dry ingredients lightly with fingertips, then stir in the dry ingredients into the banana mixture just until combined.
Scoop batter evenly into prepared muffin tins, about ¾ full. Yield may vary depending on the size of your muffin tins.
Bake until a toothpick inserted in center of a muffin comes out clean and muffins are golden brown, 14 to 16 minutes. Transfer muffin tins to a wire rack and let cool 5 minutes. Remove muffins from tin and cool on wire rack, or better yet, enjoy warm.
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Notes
Since bananas vary widely in size, it's best to measure the amount of mashed bananas so you get an accurate measurement.
If desired, make jumbo muffins or mini muffins with this batter but adjust baking time accordingly.
If you prefer, substitute all-purpose flour for the whole wheat flour, or vice versa.