Set oven to 375°F (190°C) and line a baking sheet with parchment paper.
In a large bowl, whisk flour, sugar, baking powder, cinnamon, nutmeg, and salt.
In another bowl, mix milk, egg, melted butter, and vanilla until smooth.
Pour wet mixture into dry ingredients and stir just until combined. Fold in diced apples. Batter will be thick.
Scoop ¼-cup portions of batter onto the baking sheet, spaced 2 inches apart.Bake for 15–18 minutes, until golden and a toothpick inserted comes out clean. For glaze: Let fritters cool slightly. Whisk powdered sugar, milk or cider, and vanilla. Drizzle over fritters.
For cinnamon sugar: While warm, brush with melted butter and roll in cinnamon sugar.