2. Beat the butter and sugar together with an electric mixer on high speed until the mixture is light in color and fluffy.
3. Lower the speed to low and add the eggs one at a time. Add the vanilla.
4. In a separate bowl, combine the flours, baking powder, baking soda and salt.
5. Fold half of the dry ingredients into the butter mixture using a rubber spatula or wooden spoon. Stir in the applesauce and sour cream. Fold in the remaining dry ingredients.
6. Fill the muffin tins halfway with batter. Add a few apple slices to each muffin and spoon the remaining batter over the apples.
7. Bake on the center rack until the muffins are golden brown and springy to the touch, about 35 to 40 minutes.