In a large bowl, whisk together the flour, sugar, baking powder, cinnamon, nutmeg, and salt.
In another bowl, whisk the eggs, milk, vanilla, and melted butter until smooth.
Pour the wet ingredients into the dry ingredients and stir just until combined.
Fold in the diced apples until evenly distributed throughout the batter.
Heat about 2 inches of vegetable oil in a heavy pot to 350°F (175°C).
Using a large spoon or ice cream scoop, carefully drop portions of batter into the hot oil.
Fry for 2 to 3 minutes per side until the fritters are deep golden brown and cooked through.
Transfer the fritters to a paper towel-lined plate to drain.
Whisk together the powdered sugar, milk, and vanilla until smooth.
Drizzle or brush the glaze over the warm fritters and allow it to set for a few minutes before serving.