Preheat oven to 375 degrees F (190 degrees C). Grease a 12-cup muffin tin with oil or line cups with paper liners.
Sift all-purpose flour, self-rising flour, baking powder, cinnamon, nutmeg, and salt together in a small bowl. Sift the flour mixture again into a larger bowl. Stir in brown sugar and white sugar. Make a well in the centre of the flour mixture.
Place chopped apples in a saucepan and cover with water. Cover saucepan with a lid and bring to a simmer over medium heat. Simmer until apples are just tender, but not soft, about 5 minutes. Remove from heat, drain, and set aside.
Lightly beat eggs with the vanilla extract in a bowl. Whisk in milk until well combined. Whisk in vegetable oil in a slow stream.
Pour egg mixture into the well of the flour mixture in a slow stream, stirring to combine using a metal spoon. Stir the mixture until just combined. Fold in apples. Spoon batter into the prepared muffin cups, filling each 3/4 full.
Bake in the preheated oven until tops spring back when lightly pressed and a skewer inserted in the centre comes out clean, 15 to 20 minutes.
Allow muffins to stand in the pan for 3 to 5 minutes before turning them out on to a wire rack to cool.