Preheat oven to 375 degrees F (190 degrees C). Grease a muffin pan or line with paper muffin liners.
In a large mixing bowl, cream butter and brown sugar together until fluffy. Add bananas, milk, vanilla and eggs; mix well. Stir in flour, bran, baking powder, soda and salt; blend just until moistened. Stir in walnuts. Pour batter into prepared muffin cups.
Bake at 375 degrees F (190 degrees C) for 20 to 25 minutes or until a toothpick inserted into the center of a muffin comes out clean. Cool in the cups for 5 minutes, then remove muffins and place on a wire rack to cool completely.