Preheat oven to 350 degrees. Spray donut pan with nonstick cooking spray and set aside.
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 donut cavities in pan and bake 10-12 minutes. Cool 5 minutes, then invert pan and remove. Set aside to cool 20 minutes.
For frosting, whip butter, cream cheese, powdered sugar and vanilla together until smooth.
Once donuts are cool, frost with cream cheese frosting. Serve.