In a small shallow bowl, combine sugar and cinnamon; set aside. In a mixing bowl, whisk together the all-purpose flour, baking powder, ground cinnamon, freshly grated nutmeg, and salt until well combined.

In a separate bowl, beat the eggs until smooth. Add the melted butter, vanilla extract, and apple cider to the beaten eggs. Mix well until all the ingredients are thoroughly combined. Gradually add the wet mixture to the dry ingredients, stirring until just combined. Be careful not to overmix the batter, which can result in tougher doughnuts.

In a medium deep pot or Dutch oven, heat 3 inches of frying oil to a temperature of around 350°F (175°C). Use a candy thermometer to monitor the oil temperature. Drop spoonfuls of the doughnut batter into the hot oil using a small ice cream scoop or two spoons. Fry the doughnuts in small batches, ensuring they have enough space to expand.

Cook each side for approximately 3-4 minutes or until golden brown. Fry, turning until golden brown. Using a slotted spoon or spider strainer, carefully remove the fried doughnuts from the oil and place them on a paper towel-lined plate to drain excess oil; then, soon after, dredge them in the cinnamon-sugar mixture to coat them evenly.

Alternatively, you can place the cinnamon sugar mixture in a paper bag and gently toss the doughnuts until coated. Repeat the frying and coating process with the remaining doughnut batter. Once all the doughnuts are coated, they are ready to be served. Enjoy them warm or at room temperature.

Note: Exercise caution when working with hot oil, and always follow safety guidelines for frying. Adjust the cooking time as needed based on the size of the doughnuts and the temperature of the oil.