Preheat the oven to 350°. Lightly oil ten cavities of a twelve cavity doughnut pan or a couple of six cavity doughnut pans.
To make the batter, first whisk the flour, sugar brown sugar, baking powder, baking soda, cinnamon, nutmeg, cloves, and salt together in a medium mixing bowl.
Stir the apple cider, oil, and vanilla together in a separate bowl or large liquid measuring cup.
Add the cider mixture to the flour mixture. Whisk or stir the ingredients together just until fully mixed.
Spoon (or pipe, using a pastry bag) the batter into the oiled doughnut cavities, filling each cavity to just below the brim.
Bake the doughnuts for about 12 to 14 minutes, until they spring back when lightly touched.
Transfer the doughnut pan to a cooling rack and allow the doughnuts to cool.
While the doughnuts cool, stir the sugar and cinnamon for the coating together in a small bowl.
Once cool, brush each doughnut with melted butter, then dip it in the cinnamon sugar mixture, gently rolling it around until fully coated.
Serve.