Combine the yeast and warm cider in a small bowl; set aside until foamy, about 5 minutes. In a stand mixer bowl, combine melted butter and sugar, mix well. Whisk in egg, egg yolk, milk, salt, nutmeg, cinnamon, yeast mixture, and 3 cups flour, beat until smooth. Add remaining flour until a soft ball of dough forms around the hook. Transfer dough to a greased bowl, cover, and let rise in a warm area until tripled in size, about 1 hour. Punch the dough down, turn it out onto a floured surface, and knead lightly. Roll the dough into a large rectangle about 1/2 inch thick and cut out circles with a dough cutter. Place cut doughnuts on a greased baking sheet, cover, and let rise for 45 minutes to 1 hour. Heat oil in a large pot to 360°F. Fry doughnuts for 1-2 minutes on each side until golden brown. Roll warm doughnuts in cinnamon sugar and set aside. For apple pie filling, melt butter in a skillet, add apples, salt, and sugars; cook until softened. Stir in cinnamon, apple cider, and flour, then cook until thickened. Fill each doughnut with apple pie filling using a pastry bag.