In a large bowl combine 2 cups of flour, baking powder, cinnamon, nutmeg, and salt.
Combine milk and melted butter in a small bowl, set aside.
In a stand mixer or large mixing bowl combine eggs, sugar, and vanilla for 3 minutes. Slowly alternate adding flour and milk mixture to egg mixture.
Stir in 1 cup of flour and apples.Cover and chill dough for 3 hours.
Roll out dough on floured surface to 1/2 inch thick. Note: dough can be very sticky, add 1/2 cup of flour if needed. Cut dough with doughnut cutter, dipping into flour between cuts.
Heat oil to 375 degrees.
Fry 2-3 doughnuts at a time, frying 1 minute on each side.
If using cinnamon sugar topping dip into sugar then transfer to wire rack to cool.
Drain and cool on paper towels if glazing.
If glazing, wait until doughnuts are completely cooled then dip into glazing or drizzle on top.