In a large mixing bowl, combine all-purpose flour, granulated sugar, active dry yeast, and salt. In a separate bowl, whisk together the eggs, warm milk, and melted butter. Gradually mix the wet ingredients into the dry until a dough forms. Knead on a floured surface for about 8-10 minutes until smooth and elastic. Let the dough rise in a warm place for 1-2 hours, covered, until doubled in size.

While the dough rises, prepare your desired fillings. For pumpkin cheesecake, mix pumpkin puree, cream cheese, and spices in a bowl until smooth. For chocolate pudding, prepare according to package instructions. Once thickened, refrigerate your fillings to ensure they set properly.

Once the dough has risen, gently punch it down to release the air. Roll out the dough on a floured surface until about ½-inch thick. Use a round cutter to cut circles, and a smaller cutter to create the inner hole for filled doughnuts. Place the shaped doughnuts on a baking sheet lined with parchment paper and cover with a cloth. Allow them to rise for another 30 minutes until puffy.

Heat oil in a deep pot or fryer to 350°F (175°C). Carefully drop a few doughnuts into the hot oil, being sure not to crowd the pot. Fry each side for about 2-3 minutes until golden brown, using a slotted spoon to turn them halfway. Once done, remove the doughnuts and place them on a wire rack lined with paper towels to absorb excess oil.

While the doughnuts are still warm, prepare a glaze by mixing powdered sugar with reduced apple cider and heavy cream to desired consistency. Alternatively, dip the doughnuts into cinnamon sugar for a classic finish. Coat each doughnut thoroughly and set them on the wire rack to allow the glaze to set slightly.