Heat oil: Pour oil into a heavy-bottomed pot (about 2 inches deep) and heat to 350°F (175°C).
Mix dry ingredients: In a large bowl, whisk flour, sugar, baking powder, salt, cinnamon, and nutmeg.
Mix wet ingredients: In another bowl, beat eggs, milk, melted butter, and vanilla.
Combine: Stir wet into dry until just combined. Fold in chopped apples.
Fry fritters: Carefully drop large spoonfuls of batter into hot oil, a few at a time. Flatten slightly with spoon. Fry 2–3 minutes per side until deep golden brown.
Drain: Transfer to a paper towel-lined rack to cool slightly.
Glaze: Whisk glaze ingredients until smooth. Dip each fritter or drizzle generously while still warm.