In a medium bowl, combine buttermilk, vegetable oil, and vanilla. Whisk in the eggs, then mix in the sugar and brown sugar.
In another bowl, sift flour and combine with baking powder and salt.
Spoon or pipe mixture into well greased donut pan. See notes about this
Bake for 13-15 minutes or until donuts spring back when pressed.
While donuts are baking, combine sugar and cinnamon in a shallow bowl.
When the donuts are done and still warm, use a pastry brush to coat donuts with melted butter and immediately dip donuts in cinnamon-sugar mixture until well coated.
Enjoy--preferably with a cup of hot cider or coffee!
Notes
I STRONGLY recommend using Baker's Joy (i.e. what you should use to grease a bundt pan) to grease your donut pan to prevent sticking. You can also make DIY Baker's Joy with crisco, vegetable oil, and flour.