In a large bowl, stir brown sugar and oil together. In a separate bowl, stir milk, vinegar and vanilla together and set aside. In another bowl, stir flour, baking soda, baking powder, salt, cinnamon and nutmeg together until combined. Alternate whisking wet and dry ingredients into the sugar and oil mixture. Once batter forms and all ingredients are incorporated, stir in carrot. Divide evenly between 8 doughnut cavities in pan and bake 10-12 minutes. Cool 5 minutes, then invert pan and remove. Set aside to cool 20 minutes.