Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Whisk together all-purpose flour, brown sugar, baking powder, cinnamon, nutmeg, and salt in a large bowl. Cut in butter with a pastry blender until mixture resembles coarse crumbs.
Whisk milk, egg, and vanilla extract together in a cup. Stir into flour mixture just until combined; stir in chopped apple. Use a scoop or spoon to drop 1/4-cup mounds onto prepared baking sheet.
Bake in the preheated oven until a skewer inserted near the center of fritter comes out clean, 11 to 12 minutes. Remove to a wire rack.
Preheat the oven’s broiler. Whisk together confectioner's sugar, milk, and vanilla extract. Brush a thin layer of glaze over fritters, and place under broiler.
Broil fritters until glaze is bubbly, about 2 minutes.