Preheat the oven to 350°F (175°C) and lightly grease a mini muffin pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a separate bowl, whisk together the sugar, melted butter, egg, vanilla, apple cider, and buttermilk.
Gradually add the dry ingredients to the wet ingredients and stir just until combined.
Spoon the batter into the prepared mini muffin cups, filling each about three-quarters full.
Bake for 12–15 minutes, or until the doughnut holes are lightly golden and a toothpick inserted in the center comes out clean.
Allow them to cool in the pan for a few minutes before transferring them to a wire rack.
Combine the granulated sugar and cinnamon in a shallow bowl.
Brush each warm doughnut hole lightly with melted butter.
Roll the doughnut holes in the cinnamon sugar until evenly coated.
Serve warm or at room temperature.