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

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

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

Fit a hand mixer with the whisk attachments and beat egg whites until soft peaks form. Gently fold whites into batter until completely blended.

In a medium bowl, combine remaining 1/2 cup sugar with remaining 1 teaspoon cinnamon.

Add enough oil to a heavy-bottomed Dutch oven or a deep cast iron skillet to come halfway up sides of pan. Heat oil over medium-high heat to 350 degrees. When oil is hot, carefully add 2 tablespoonful-scoops of batter, working in batches and being careful not to overcrowd pan. Maintain an oil temperature of between 300 and 350 degrees by adjusting the burner as necessary. Fry fritters in small batches until golden and cooked through, about 4 to 5 minutes, turning them as needed for even color. Remove fritters with a slotted spoon or strainer, drain briefly, then toss them in the cinnamon sugar. Transfer fritters to a serving platter. Repeat with remaining batter. Right before serving, roll fritters a final time in cinnamon sugar.