Preheat oven to 350°F (175°C) and grease two 6-cavity donut pans.
In a large bowl, whisk together flour, sugars, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a separate bowl, whisk together buttermilk, eggs, melted butter, and vanilla extract.
Fold the wet ingredients into the dry ingredients until just combined.
Gently fold in the finely diced apples.
Pipe the batter into the prepared donut pans, filling each cavity approximately 3/4 full.
Bake for 12 to 15 minutes or until the tops spring back when lightly touched.
Cool donuts in the pan for 5 minutes before transferring to a wire rack.
In a small bowl, whisk together powdered sugar, milk, and 1/2 teaspoon vanilla extract to create the glaze.
Dip the tops of the cooled donuts into the glaze and place on a wire rack until set.