Preheat your oven to 375°F (190°C). Grease a muffin tin or line with paper muffin liners.
In a large bowl, mix together wheat bran and milk. Let stand for 10 minutes.
In another bowl, beat together applesauce, egg, oil, brown sugar, and vanilla extract.
Add the applesauce mixture to the wheat bran mixture and stir until combined.
In a separate bowl, whisk together flour, baking soda, baking powder, and salt. Add to the bran mixture and stir until just blended.
Spoon batter into prepared muffin tin, filling each cup about ⅔ full.
Bake in preheated oven for 15-20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Cool in the tin for 5 minutes, then transfer to a wire rack to cool completely.