Heat the oven to 375 degrees. Line a 12-cup muffin tin with bake cups and set aside.
In a small microwave-safe bowl, melt the butter on high for about 30 to 60 seconds; set aside to cool slightly.
Sift together the flour, baking powder, baking soda, cinnamon, and salt into a large mixing bowl. When mixing dry and wet ingredients for muffins, gently stir just until the flour disappears. This will produce muffins with a nice crumbly texture.
In another large bowl, whisk together the eggs and brown sugar. Stir in the applesauce and melted butter until the mixture is smooth. For mess-free muffins, use an ice-cream scoop to spoon the batter into the bake cups.
Pour the apple mixture over the flour mixture. Mix with a wooden spoon until combined (it's ready when you can't see any traces of flour).
Fill the bake cups about two-thirds full with batter. Bake for 20 minutes or until light brown.