Preheat the mini donut maker. If using an oven preheat to 350ºF and grease a mini donut pan.
Prepare the batter. In a mixing bowl, combine apple sauce, apple cider, and eggs. Stir until eggs are fully incorporated.
Mix dry ingredients. Add sugar, melted butter, flour, baking powder, salt, and vanilla extract to the wet ingredients. Stir gently until just combined - don't overmix! A few lumps in the batter are fine.
Fill the donut maker or pan. Transfer the batter into a piping bag or a zip-top bag with the corner cut off. Pipe the batter into the mini donut molds in one quick motion. Avoid overfilling.
Cook the donuts. If using a mini donut maker, cook for 8-11 minutes, or until golden brown. If baking in the oven, bake for 8-14 minutes until golden.
Coat the donuts. Melt 2 tablespoons of butter in a zip-top bag. Add the freshly baked donuts and shake to coat. In a separate bowl, mix sugar and cinnamon, then add the mixture to the bag and shake to coat the donuts evenly.