1. Preheat the oven to 350 degrees Fahrenheit. Grease two doughnut pans.
2. Stir flour, sugar, salt, baking powder, cinnamon, baking soda, nutmeg, and cloves in a mixing bowl. Add egg, apple cider, applesauce, oil, and vanilla and whisk until smooth. Transfer the batter to a piping bag or Ziploc bag. Snip the corner to form a small hole and pipe the batter into the pans, until just about filled. Bake for 12-15 minutes (or three minutes in a preheated doughnut machine). Transfer to a cooling rack.
3. Mix sugar and cinnamon in a small bowl until combined. Use a pastry brush to brush doughnuts with melted butter, then toss in cinnamon-sugar while still hot. Let cool.