Preheat oven to 375°F. Line a 12-cup muffin pan with liners and lightly mist with cooking spray.
Whisk flour, baking powder, baking soda, cinnamon, nutmeg, and salt in a large bowl.
In another bowl, whisk granulated sugar, brown sugar, applesauce, oil, and vanilla until smooth.
Pour in buttermilk and whisk just to combine the wet mixture.
Add wet ingredients to the dry bowl. Stir gently until only a few streaks of flour remain.
Fold in diced apples and walnuts until evenly dispersed. Do not overmix.
Divide batter among muffin cups. Sprinkle tops with turbinado sugar.
Bake 18–22 minutes, until domed and a tester comes out clean. Cool 5 minutes, then transfer to a rack.