In a large mixing bowl, whisk together the sifted flour, baking powder, baking soda, salt, cinnamon, and nutmeg until well combined. This ensures even distribution of the leavening agents and spices.

In another bowl, beat the egg and mix in the granulated sugar until well combined. This will help to aerate the mixture, creating a lighter doughnut.

Add the buttermilk and vanilla extract to the egg mixture, whisking until smooth.

Gradually add the wet ingredients to the dry ingredients, stirring gently until just combined. Be careful not to overmix, as this can lead to tough doughnuts.

Fold in the finely diced apple, ensuring it is evenly distributed throughout the batter.

In a deep frying pan or deep fryer, heat the vegetable oil over medium heat until it reaches 350°F (175°C). Use a candy thermometer to monitor the temperature, as maintaining the right heat is crucial for frying.

Using a spoon or a small ice cream scoop, carefully drop spoonfuls of the batter into the hot oil, being careful not to overcrowd the pan.

Fry the doughnuts for about 2-3 minutes on each side, or until they are golden brown. Use a slotted spoon to turn them, ensuring even cooking.

Once cooked, remove the doughnuts from the oil and place them on a cooling rack lined with paper towels to drain excess oil.

In a small bowl, whisk together the sifted powdered sugar, milk, and vanilla extract until smooth. Adjust the milk quantity to achieve your desired glaze consistency.

Once the doughnuts are slightly cooled, dip the tops into the glaze, allowing any excess to drip off. Place the glazed doughnuts back on the cooling rack to set.