Preheat oven to 400 degrees F (200 degrees C). Grease a 12-cup muffin pan or line with paper liners.
Stir 1/4 cup oats, 1 tablespoon brown sugar, melted butter, and 1/4 teaspoon cinnamon together in a small bowl; set aside.
Whisk together 1 1/2 cups oats, flour, 1/2 cup brown sugar, 1 1/2 teaspoons cinnamon, baking powder, baking soda, and salt together in a large bowl.
Stir applesauce, milk, oil, egg, and vanilla extract together in a bowl. Stir applesauce mixture into flour mixture just until all ingredients are moistened; stir in apple. Spoon apple mixture into prepared muffin cups about 2/3 full; sprinkle oat topping mixture evenly over each muffin.
Bake in the preheated oven until a toothpick inserted near the center comes out clean, about 15 minutes.