Heat oven to 400 degrees F. Fill 18 indents of 2 muffin pans with liners; set aside.
In a small bowl, whisk together flours, baking soda, cinnamon, salt and nutmeg. In another small bowl, stir together 1/4 cup brown sugar and half of the apple pieces; set aside.
Beat butter and remaining 3/4 cup brown sugar together on medium-high speed until well blended, about 2 minutes. Add eggs, one at a time, beating well after each. Beat in applesauce and vanilla extract.
Reduce speed to low and alternately add flour mixture and buttermilk, starting and ending with flour. Stir in remaining apple pieces.
Fill each muffin liner with 1/4 cup batter. Sprinkle 1 tablespoon apple-brown sugar mixture on top of each muffin. Bake muffins at 400 degrees F for about 20 minutes or until toothpick inserted in center comes out clean. Cool on a rack for 15 minutes, then remove from pan and allow to cool completely.