For donuts: Preheat oven to 350 degrees. Add 1 Tbsp. butter to a small pot along with minced apple. Cook over medium heat until soft, about 5 min. Meanwhile, combine dry ingredients in a large bowl. Whisk wet ingredients (including the remaining 1 Tbsp. melted butter) together in a separate bowl. Add wet ingredients to dry ingredients and whisk until a thick batter forms. Fold in cooked apples. Spoon batter into a greased 6-cavity doughnut pan, filling each mold about three-quarters of the way full. Bake for 10 min., or until a toothpick comes out clean. Let doughnuts cool completely before glazing.
For glaze: Whisk milk and powdered sugar until mixture is smooth. Drizzle over donuts or gently dip the top of each one into the glaze. For a thicker coating, allow the glaze to dry and dip again. Sprinkle generously with Spiced Vanilla Bean Sugar.