Melt butter in a skillet over medium heat. Add the chopped and peeled apples, brown sugar, and cinnamon. Cook 4 to 5 minutes, stirring, until just softened and most of the liquid has cooked off. Spread on a plate and cool 10 minutes.
Heat 2 inches of oil in a heavy pot to 350°F.
Whisk powdered sugar, apple cider, and vanilla until smooth. Set aside.
In a large bowl, whisk discard, egg, milk, brown sugar, and vanilla. In another bowl, whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Stir dry into wet just until combined. Fold in cooled apples.
Scoop ¼ cup portions into the oil and gently flatten to about ½ inch thick. Fry 3 to 4 at a time, 2 to 3 minutes per side, until deep golden.
Transfer to a wire rack or paper towels.
Dip or drizzle the warm fritters with glaze. Let set 5 minutes and serve.