Preheat your oven to 180°C (350°F) and grease or line a muffin tin with paper liners.
In a large mixing bowl, whisk together the eggs and brown sugar until the mixture is smooth and slightly frothy, about 2 minutes. The sugar should be partially dissolved with the eggs' color brightening.
Add the molasses and vegetable oil to the egg mixture, stirring until the ingredients are well combined and the mixture is uniform in color.
In a separate sifter or bowl, combine the all-purpose flour, baking soda, and whole wheat bran. Mix gently to distribute evenly.
Gradually pour the dry ingredients into the wet mixture, folding gently with a spatula or spoon. Stop once just combined; the batter should be slightly lumpy but well integrated.
Slowly add the milk to the batter, folding gently until the mixture reaches a thick but pourable consistency. If it feels too thick, add a splash more milk, one teaspoon at a time.
Spoon the batter evenly into the prepared muffin cups, filling each about three-quarters full. Tap the tin lightly on the counter to settle the batter and even out the tops.
Bake in the preheated oven for about 20-25 minutes, or until the muffins turn a golden brown and a toothpick inserted into the center comes out clean. The muffins should have risen and feel firm but springy.
Remove the muffins from the oven and let them cool in the tin for about 5 minutes. Then transfer them to a cooling rack to cool slightly and develop their full flavor and texture.
Serve the muffins warm or at room temperature, plain or with a pat of butter. Enjoy the comforting, nutty aroma and moist, chewy crumb with each bite.
