Preheat the oven to 375 degrees F (190 degrees C). Grease a muffin pan or line with paper liners.

Cream together butter and brown sugar in a large bowl with an electric mixer on medium speed until fluffy. Add mashed bananas, eggs, milk, and vanilla; mix well. Stir in flour, bran, baking powder, baking soda, and salt; blend just until moistened. Stir in walnuts. Pour batter into the prepared muffin cups.

Bake in the preheated oven until a toothpick inserted into the center of a muffin comes out clean, 20 to 25 minutes. Cool muffins in the pan for 5 minutes, then transfer to a wire rack to cool completely.