In a large bowl, mix together the yeast, 2 tablespoons of the sugar and the water to make a slurry. Let sit aside for five minutes until yeast is bubbling.
In a separate bowl, mix together flour, 2 tablespoons sugar, salt, cinnamon, nutmeg, vanilla and egg yolks, then add to the yeast slurry. Knead the dough until it forms a ball. Then add the butter and knead to incorporate the butter. Cover and let rise in a warm place for an hour until it doubles in size.
Meanwhile, make the filling. Melt butter in a large saute pan over medium heat. Add the apples and saute until soft, about five minutes. Then add sugar, lemon juice, flour and cinnamon and stir to combine. Bring mixture to a boil, then lower heat and cook for two minutes until thick. Let cool.
After proofing, roll out the dough to 1/8-inch thickness. Cut into 1 1/2- or 2-inch rounds with a cookie cutter or glass and put dough rounds on a cookie sheet. Cover with plastic wrap and let dough relax for 10 minutes. Repeat with excess dough.
Flatten out half of the doughnut circles slightly and place a rounded teaspoon of the apple filling in the center of each circle. Place the remaining circles of dough over the filling, and press down the edges well to seal. Heat two to three inches of oil in a heavy-bottomed Dutch oven or deep fryer to 350 degrees.
Fry doughnuts a few at a time until cooked through and golden brown for one to two minutes, and then flip and fry the other side. Remove from the oil with a slotted spoon and drain on paper towels and dip into cinnamon sugar mixture. Serve immediately.