Preheat the oven to 350 degrees F (175 degrees C). Grease two 12-cavity mini doughnut pans.
Combine flour, sugar, baking powder, salt, nutmeg, and cinnamon in a large bowl; mix together with a fork until combined. Add shortening and mix with the fork until broken up into small clumps.
Mix together apples, milk, and eggs in a separate bowl. Add to flour mixture and stir just until blended. Spoon batter into the prepared doughnut pans.
Bake doughnuts in the preheated oven until slightly browned on the bottoms, about 10 minutes. Remove from pan and transfer to a wire rack to cool completely.
While doughnuts are cooling, place melted butter for topping in a shallow bowl. Mix together 1 cup sugar and 2 teaspoons cinnamon in another shallow bowl. Dip cooled doughnuts in butter and then in sugar mixture to coat.