Peel, core and chop the apples. Sprinkle with about 1 tbs lemon juice, if desired, to stop them browning.
Sift the flour, baking powder, salt, cinnamon and nutmeg into a large bowl. Whisk in the sugar. Whisk in the eggs, milk and vanilla. Stir in the melted butter and the apples
Add enough oil in a dutch oven so that it’s about 1 1/2-2-inches/3-5 cm up the sides.
Heat over medium heat until oil temperature reaches 350F/180C
Drop the batter by tablespoonsful into the hot oil; flattening slightly in the center. Fry only about 3 at a time. Fry until the donut is dark brown, turn and fry until the other side is dark brown—about 4 minutes per side.
Place cooked donuts on a plate lined with paper towels. Repeat with remaining batter.
Mix the cinnamon and sugar. Roll warm donuts in the cinnamon sugar until well coated.