Preheat oven to 425°F. Grease 14 standard (2½-inch) muffin cups with nonstick cooking spray.
Whisk buttermilk, eggs, oil, and vanilla extract together in large mixing bowl until well-combined.
Stir in carrot until combined.
Add all-purpose flour, whole wheat flour, oats, granulated sugar, brown sugar, baking powder, cinnamon, baking soda and salt; stir just until combined. Don't overmix.
Spoon batter evenly into prepared muffin tin cups.
Sprinkle with coarse sugar, if using.
Bake 5 minutes.
Without removing the muffins, decrease oven temperature to 350F and continue baking for 15-20 minutes or until the muffins spring back when lightly pressed or a toothpick inserted into centers comes out clean.
Cool in the pan for 5 minutes.
Remove to a wire rack to finish cooling.
Notes
Buttermilk: Regular buttermilk or low-fat buttermilk will work. If you don't have buttermilk, you can make a homemade buttermilk substitute with this recipe.
Vegetable oil: Or a similar neutral oil.
Brown sugar: Light brown sugar or dark brown sugar will work. I don't pack the brown sugar, but you can pack it if you'd like sweeter muffins.