Preheat the oven to 350°F and grease your donut pan. While the oven is heating, mix together your egg, butter, and sugars in a large mixing bowl. Once those are mixed and smooth, add in the buttermilk, vanilla extract, and apple cider. Blend thoroughly. Sift the remaining dry ingredients into the wet ingredients. Whisk together gently, and be sure to not overmix! This is important to give it a chewy, doughy texture.
Pour your batter into a donut hole pan, filling each mold ¾ full. (If you add more, they will puff up and spill over, losing their round shape.) Bake for 5-6 minutes until barely golden.
While the donut holes are cooling, mix together your sugar and cinnamon for the coating. Once they have cooled, dip them in a light coating of butter, then roll in the cinnamon sugar mixture. Serve immediately (before they're all gone).