Preheat oven to 350F and lightly grease a 12-cup muffin tin with cooking spray.
In a large bowl, whisk together the flour, wheat bran, brown sugar, baking powder, salt and spices.
In a small bowl, whisk together the vegetable oil, buttermilk, egg and vanilla. Make a well in the center of the dry ingredients and pour buttermilk mixture into the well. Stir until just combined, then add in the shredded carrots and diced apples.
Divide evenly into muffin tins and bake for 16-20 minutes, until a toothpick inserted into the center comes out clean and the top of the muffin springs back when lightly pressed.
Cool muffins on a wire rack. Store in an airtight container.