Preheat oven to 350 degrees F (175 degrees C). Lightly grease twelve 2 1/2-inch muffin cups.
Stir together bananas, brown sugar, applesauce, egg, and vanilla extract in a large bowl. Sift together flour, baking powder, salt, apple pie spice, and baking soda in another bowl.
Add flour mixture and oats to banana mixture; gently stir just until combined. Do not overmix. Fold in apple. Spoon batter evenly into prepared muffin cups.
Bake in the preheated oven until a toothpick inserted into the centers comes out clean, 15 to 20 minutes. Remove muffins from pan, transfer to a wire rack, and let cool completely.
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