Preheat the oven to 400 degrees F (200 degrees C). Grease a 12-cup muffin tin.

Mix together brown sugar, applesauce, canola oil, egg, vanilla extract, and salt in a large bowl.

Mix flour, bran, flax, baking soda, and baking powder in a separate bowl. Add bran mixture to applesauce mixture. Add milk and stir until thoroughly combined. Pour batter into the prepared muffin cups.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 11 to 13 minutes.