Adjust the oven rack to the middle position and heat the oven to 200°F (93°C).
Melt 1 tablespoon of butter in a 12-inch skillet over medium heat. Add the apples, 1 tablespoon sugar, ½ teaspoon cinnamon, and 1 tablespoon lemon juice. Increase the heat to medium-high and cook, stirring often, until the liquid has evaporated and the apples look dry. Remove from heat and let cool completely.
In a large bowl, whisk together the flour, granulated sugar, baking powder, salt, 2 teaspoons cinnamon, and ½ teaspoon nutmeg.
In a separate bowl, whisk together 1½ cups of apple cider, eggs, and vanilla extract until smooth.
Add the cooled apples to the flour mixture and toss to combine. Stir in the cider mixture until fully incorporated.
Set a wire rack inside a rimmed baking sheet and line it with a triple layer of paper towels.
Add oil to a 12-inch cast-iron skillet until it measures about ½ inch deep, and heat over medium heat to 325°F (163°C).
Using two greased spoons, carefully scoop and drop five heaping mounds of batter into the oil.
Fry until deep golden brown, 6 to 8 minutes, flipping halfway through. Adjust the heat as needed to maintain the oil temperature between 300°F and 325°F.
Transfer the fritters to the prepared rack and keep warm in the oven. (Fritters can be kept warm for up to 30 minutes.) Return oil to 325°F and repeat with the remaining batter.
In a medium bowl, whisk together the confectioners’ sugar, remaining ½ teaspoon cinnamon, remaining ¼ teaspoon nutmeg, 1 tablespoon melted and cooled butter, and ¼ cup apple cider until smooth.
Drizzle 1 heaping tablespoon of glaze over each fritter. Let the glaze set for 10 minutes before serving.