Preheat the oven to 350°F. Lightly grease a donut pan or use a silicone donut pan.
In a large mixing bowl, whisk together the flour, cane sugar, brown sugar, baking powder, baking soda, salt, cinnamon and nutmeg.
In a separate bowl, whisk together the apple cider, melted butter, Greek yogurt, egg and vanilla extract.
Pour the wet ingredients into the dry ingredients. Gently stir just until combined, being careful not to overmix the batter.
Transfer the batter to a piping bag or large zip-top bag with one corner snipped off. Pipe the batter into the prepared donut cavities, filling each about three-quarters full. You could also use a spoon to fill.
Bake for 10–12 minutes, or until the tops spring back when lightly touched and a toothpick inserted into a donut comes out clean.
Let the donuts cool in the pan for 5 minutes, then carefully transfer them to a wire cooling rack.
Place the melted butter in a shallow bowl. In a separate shallow bowl, stir together the cane sugar and cinnamon.
Dip each slightly warm donut in the melted butter, then coat it generously in the cinnamon-sugar mixture. Serve warm and enjoy.