Preheat oven to 400 degrees F (200 degrees C). Grease 12 muffin cups or line with paper liners. Combine bran flakes cereal, whole wheat flour, all-purpose flour, baking powder, and salt in a bowl. Whisk applesauce, sugar, vegetable oil, and egg together in a separate bowl; stir into flour mixture until just combined. Fill muffin cups about 2/3-full with batter. Bake in the preheated oven until a toothpick inserted in the center of a muffin comes out clean, about 15 minutes. Cool in the pan for 5 minutes on a wire rack; serve warm.