Preheat the oven to 400°F and line a baking sheet with a silicone baking mat, ensuring it is broiler-safe.
In a large bowl, whisk together all-purpose flour, white whole wheat flour, brown sugar, baking powder, cinnamon, nutmeg, allspice, and salt.
Cut in butter until the mixture resembles coarse crumbs.
In a separate small bowl, whisk together milk, egg, and vanilla extract, then add this to the flour mixture and stir until just combined.
Gently fold in the chopped apple.
Drop the batter onto the prepared baking sheet in ¼ cup portions or use a heaping tablespoon for smaller fritters.
Bake for 11-13 minutes until the bottoms are golden brown and the centers are set; for smaller fritters, check at 8 minutes.
Remove the fritters to a wire rack and turn the oven to broil.
While the oven heats, prepare the glaze by whisking together confectioners’ sugar, milk, and vanilla extract until smooth.
Brush a thin layer of glaze over the fritters, then broil for 2-4 minutes until the tops turn golden and the glaze bubbles.
Carefully remove the baking sheet, transfer fritters to a wire rack, and brush with the remaining glaze.