Melt the butter in a 12-inch skillet over medium-high heat. Add the apples and sauté, stirring frequently, for 2 minutes. Sprinkle in the brown sugar, 1/4 teaspoon of the cinnamon, the nutmeg, and salt, and cook for 2 to 3 minutes more, until the apples are lightly coated with the syrup. Remove the skillet from the heat and set aside to cool.

In a medium bowl, whisk together the egg yolks and apple cider. Stir in the cooled apple mixture.

In a large bowl, combine the flour, baking powder, and the 1 tablespoon granulated sugar. Make a well in the center of the flour and add the apple mixture. Gradually incorporate the flour into the wet ingredients, mixing gently with a whisk until uniform. Set the batter aside for 20 minutes.

Preheat the vegetable oil in a deep fryer to 350°F. Set a wire rack over a paper-towel- lined baking sheet.

Using an electric hand mixer, whip the egg whites until soft peaks form. Gently fold the whites into the batter until completely blended.

In a medium bowl, combine the remaining 1 cup sugar with the remaining 3/4 teaspoon cinnamon. Set aside half of this mixture in a medium container for the final sprinkling.

Working in batches of 3 or 4, and being careful not to overcrowd the fryer, add 2-tablespoon scoops of the fritter batter to the fryer. Fry the fritters for 4 to 5 minutes, until golden and cooked through, turning them as needed for even color. Remove the fritters with a slotted spoon or strainer, drain briefly, toss them in the bowl of cinnamon sugar, and set them aside on the wire rack.

Once you have finished frying them, roll the fritters a final time in the reserved cinnamon sugar, or just sprinkle the cinnamon sugar over the top. Serve immediately.