Place the peeled bananas in a large bowl and beat them well with an electric mixer. The riper the bananas and the more you beat them, the more tender your muffins will be. There will still be some lumps, however.
In a small bowl, combine the flour, baking soda, and salt.
Add the butter, applesauce, and eggs to the banana and mix well.
Add the dry ingredients to the banana mixture and stir JUST until the batter is thoroughly combined.
Pour the batter into greased muffin cups - mine just about filled the 12 cup pan.
Bake at 350 degrees for 20-25 minutes, or until a toothpick inserted comes out clean.
Cool for 10 minutes on a rack in the pan; then remove and cool completely.