Preheat the oven to 350 degrees. Spray a regular donut pan with non-stick cooking spray.
In a large bowl, combine butter, sugar, eggs, and vanilla. Stir in applesauce, milk and yogurt. Whisk until smooth.
In a separate bowl, combine flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Add to wet ingredients and stir until fully incorporated. Fold in the apples
Divide the batter among 16 donut cavities, filling ¾ of the way full.
Bake for 12-14 minutes, until firm to the touch or passes the toothpick test (a toothpick inserted comes out clean).
Allow to cool in the pan for 10 minutes. Remove the donuts from the pan and place on a cooling rack.
While the donuts finish cooling, mix donut frosting. Using a fork, cream the butter. Add vanilla and mix until combined. Mix in applesauce, still using the fork. Whisk in the powdered sugar with the fork until smooth.
When cool, frost or dip each donut in the frosting. It is a great consistency that you can dip it like a glaze but it will hold up like a frosting.